Approximation Algorithms for Variable-Sized and Generalized Bin Covering

App ro ximation Algo rithms fo r Generalized and V a ria ble-Sized Bin Covering Matthi as Hellwi g ∗ and Al exander Souza † Abstrac t In this paper, we consider the G eneralized Bin Covering problem: W e are given m bin types, wher e each bin of type i has profit p i and demand d i . F urthermore, there a re n items, wher e item j has size s j . A bin of type i is cov er ed if the set o f items assigned to it has total size at leas t the demand d i . In that case, the profit of p i is earned and the ob jective is to maximize the total profit. T o the best of our knowledge, only the cases p i = d i = 1 ( Bin Covering ) and p i = d i ( V ariable-Sized Bin Covering ) have b een treated b efore. W e study tw o mo dels o f bin supply: In the unit supply mo del, we have exactly one bin of each type, i. e., we hav e individual bins. B y co nt rast, in the infinite supply model, we hav e arbitrarily many bins of ea ch t y pe . Clear ly , the unit supply mo del is a genera lization of the infinite supply mo del, since we can simulate the latter with the former by in tro ducing s ufficient ly many copies of each bin. T o the b est of our knowledge the unit supply mo del has not been studied yet. It is well-known that the problem in the infinite supply mo del is NP-hard, which can b e seen by a straightforw a rd reduction from P ar tition , and this ha r dness carries ov er to the unit supply mo del. This also implies that the problem can not b e approximated b etter than tw o, unless P = NP. W e b egin our study with the unit supply mo del. Our results therein hold not only asymptotically , but for all instances. This contrasts most of the previous work on Bin Cove ring , which has b een asymptotic. W e prov e that there is a c ombinatorial 5-approximation alg orithm for G eneralized Bin Covering with unit supply , which has running time O ( nm √ m + n ) . This a lso tr a nsfers to the infinite supply mo del by the ab ov e a r gument. F urthermore, for V ariable-Sized Bin Covering , in which we hav e p i = d i , we show that the na tural and fast Next Fit Decreasing ( nfd ) alg orithm is a 9  4-a ppr oximation in the unit supply mo del. The b ound is tight for the algor ithm a nd close to b eing b est-p ossible, since the pr oblem is inapproximable up to a factor of tw o, unless P = NP. Our analys is gives deta ile d insight into the limited extent to w hich the optimum can significantly outp erfo rm nfd . Then the question arises if we can improv e on those res ults in asymptotic notions, where the optimal profit div erg es. W e dis c uss the difficulty of defining a symptotics in the unit supply model. F or t wo natura l definitions, the negative result holds that V a r iabl e- Sized Bin Covering in the u nit supply mo del do es not allow a n AP T AS. Clea rly , this also excludes an APT AS for Generalized Bin Covering in that mo del. Nonetheless, we show that there is a n AFPT AS for V ariable-Sized Bin Covering in the infinite supply mo del. ∗ Humboldt Universit y of Berlin, Germany , mhel lwig@infor matik.hu- berlin.de † Universit y of F reiburg, German y , souza@info rmatik.un i-freiburg.de 1 1 Intro duction Mo dels and Motivation In this pap er, w e stud y generalizations of the NP -hard classical Bin Covering pr oblem. In this problem, w e hav e an infinite su pply of un it-sized b in s and a collecti on of items ha ving individu al s izes. Th e ob jectiv e is to pack items in to as many bins as p ossible. That is, w e seek to maximize the n u m b er of c over e d b ins, w here a b in is co v ered if the total size of the pack ed items is at least the size of the b in. T his pr ob lem is the dual of the classica l Bin P acking problem, where the goal is to pac k the items into as few bins as p ossible; see the survey [2]. Bin Covering has receiv ed considerable atten tion in the past [1, 3 – 6, 11]. W e will survey r elev ant literature b elo w. In Generalized Bin Covering , we h a ve a s et I = { 1 , . . . , m } of bin typ es and eac h b in i ∈ I of some t yp e has a pr ofit p i and demand d i . W e denote the set of items by J = { 1 , . . . , n } and define that eac h item j ∈ J has a size s j . A b in is c over e d or fil le d if the total size of the p ac ke d items is at least the demand d i of the b in, in w hic h case we earn profit p i . The goal is to maximize th e total profit gained. The s p ecial case with p i = d i is known as V ariable-S ize d Bin Covering . T o the b est of our kn owledge, the mo del w ith general profits and d emands has n ot b een s tu died in the Bin Covering s etting b efore. F ur thermore, w e consider tw o mo dels regarding the su pply of bins: In the infinite supply mo del – as the name suggests – we h a ve arbitrary many bins a v ailable of eac h bin t yp e. By con trast, we in tro duce the unit su pply mo del , in which w e ha ve one bin p er t yp e a v ailable, i. e., we sp eak of individual bins rather than bin t y p es. Observe that the unit supp ly mo d el is more general than the infi nite supp ly mo del: By in tro ducing n copies of eac h bin, we can sim u late the infinite supp ly mo del with the unit supp ly mo d el. The con verse is obvio u sly not true. F or motiv ating these generalizatio ns, we men tion the follo wing t wo applications from truc k- ing and canning. In the first app lication, supp ose that a m oving company receiv es a co llection of inquiries for mo ving contrac ts. Eac h in quiry h as a ce rtain volume and yields a certain profit if it is serve d (en tirely). The company h as a fleet of trucks, where eac h tr u c k h as a certain capacit y . The ob jectiv e is to decide w hic h inquiries to serv e with the av ailable truc ks as to maximize total profit. This problem clearly m aps to Generalized Bin Covering in the unit sup ply mo d el: the inquiries relate to the bins, while the truc ks relate to th e items. Notice that the u nit su p ply mo d el is essen tial here, since there the inquiries are in d ividual, i. e., are not a v ailable arb itrarily often. Notic e that all pr evious work on Bin Covering exclusively considers the infinite su pply mo del and is hen ce n ot applicable here. Also notice that the Generalized Bin Covering prob lem applies esp eciall y if the pr ofits d o not necessarily correlate with the volume, but also dep end on the types of go o ds. F or example, ship p ing a smaller amount of v aluables ma y yield higher profi t th an shippin g a larger amount of b ooks. As a second app lication, consider a canning factory , in w h ic h ob jects, e.g., fish, hav e to b e pac ke d int o bins (of certain t yp es ha vin g different sizes), suc h that the total pac ked w eight reac hes at least a certain resp ectiv e threshold v alue. Here it is reasonable to assu m e that the a v ailable num b er of bins is arbitrary , i. e., the infinite su pply mo del is suitable. If th e p rofits are pr op ortional to the threshold v alues, then w e hav e an application for the V ariable-Sized Bin Co vering problem. Let I denote the f amily of all bin type sets and J the family of all item sets. F urthermore, let alg ( I , J ) and o p t ( I , J ) b e the resp ectiv e p rofits gained b y some algorithm alg an d b y an optimal algorithm opt on an instance ( I , J ) ∈ I × J . The appr oximation r atio of an 2 algorithm a lg , is defined by ρ ( alg ) = sup { opt ( I , J ) alg ( I , J )  I ∈ I , J ∈ J } . I f ρ ( alg ) ≤ ρ holds for an algorithm al g with ru nning time p olynomial in the input size, then it is called a ρ - appr oximation . If there is a ( 1 + ε ) -approximat ion for ev ery ε > 0, then th e r esp ectiv e family of algorithms is called a p olynomial time appr oximation scheme (PT AS). If the run - ning of a PT AS is additionally p olynomial in 1  ε , then it is called a ful ly p olynomial time appr oximation scheme (FPT AS). With ¯ ρ ( alg ) = lim p →∞ sup { opt ( I , J ) alg ( I , J )  I ∈ I , J ∈ J , o pt ( I , J ) ≥ p } we denote the asymptotic appr oximation r atio of an algorithm alg . The notions of an asymptotic appr o ximation algorithm and of asymp totic (F)PT AS (A(F)PT AS ) transfer analogously . Our Contribution In terms of results, we mak e the follo wing con trib utions. In Section 2 we consider Gener alized Bin Covering in the unit su pply mo del. Our fi rst main r esult is a 5-appro ximation algorithm with running time O ( nm √ m + n ) in Theorem 1 . T he basic idea is to define an algorithm for a mo d ified v ersion of the problem. Even though th is solution ma y not b e feasible for the original problem, it will enable us to p ro vid e a go o d solution for the original pr oblem. As a side r esult, which might b e in teresting in its o wn righ t, we obtain an in tegralit y gap of t wo for a linear program of the mo d ified p roblem and a corresp onding in teger linear program. F or V ariable-Sized Bin Covering in the infinite supply mod el, it is not hard to see that an y reasonable algorithm (using only the largest bin t yp e) is an asymp totic 2-appro ximation. The situation changes considerably in the u ni t supply mo del: Firstly , limita tions in bin a v ailabilit y hav e to b e r esp ected. S econdly , the desired appro ximation guarantee s are n on- asymptotic (where we explain the issue concerning the asymptotics b efore T heorem 25). Our main r esu lt here is a tight analysis of the Next Fit Decreas ing ( nfd ) algorithm in the u nit supply mo del for V ariable-Sized Bin Covering , w h ic h can b e found in S ection 3. Theo- rem 8 states that nfd yields an approxima tion ratio of at most 9  4 = 2 . 2 5 w ith runn ing time O ( n log n + m log m ) . The app ro ximation guaran tee is tight for the algorithm, see Example 7 . The main idea b ehind our analysis is to classify bins according to th eir co verage : The bins that nfd co v ers with single items are – in some sense – optimally co vered. I f a b in is co v ered with at least t wo items, then their total size is at most twice the demand of the co ve r ed bin. Hence those bins yield at least half the ac h iev able profit. Intuitiv ely , the problematic bins are those that are n ot co vered by nf d : An optimal algorithm might recom b ine lefto v er items of nfd with other items to co v er some of these bins and in crease the p rofit gained. Our analysis giv es insight int o the limited extend to wh ic h su c h recom binations can b e p rofitable. Firstly , our result is interesting in its own righ t, s ince nfd is a natural and fast algorithm. Secondly , it is also close to b eing b est p ossible, in the follo wing sense. A folklo re reduction fr om P ar tition yields th at ev en the classical Bin Covering pr oblem is not appro ximable within a factor of t wo , unless P = NP . This clearly excludes the p ossibilit y of a PT AS for Bin Covering in any of the mo d els. The reduction crucially us es that there are only tw o iden tical bins in the Bin Covering instance it creates. Then the qu estion arises if one ca n impro ve in an asymptotic notion, wh ere the op timal profit dive r ges. Indeed, for the classical Bin Covering p r oblem with infinite supply , there actually is an A(F)PT AS [5, 11]. Ho we v er, since w e h a ve individual bins rather than bin typ es in the unit supply m o del, there are difficulties f or d efining a meaningful asymp totics for V ariable-Sized Bin Covering 3 therein. W e discuss this issue in more detail b efore T heorem 25. Moreo v er, in Th eorem 25 w e sho w that, ev en if there are m > 2 b ins av ailable and the optimal profit diverges, there are instances, for whic h no algorithm can h av e an app ro ximation ratio smaller than 2 − ε for an asymp toticall y v anishing ε > 0, unless P = NP . Intuitiv ely , we show that, ev en in this asymptotic notion, one still has to solv e a P ar tition instance on t wo “la rge” bins. Hence, f or this asymptotics, there is no APT AS for V ariable-Sized Bin Covering in the unit s upply mo del, u nless P = NP . Ho we v er, this fact do es not exclude the p ossibilit y of an A(F)PT AS for V ariable-Sized Bin Covering in the infinite su pply mo del. Indeed, w e can giv e an A(F)PT AS for V ariable-Size d Bin Co vering w ith infinite su pply . Our algorithm is an extension of the APT AS of Csir ik et al. [5] for classical Bin Covering . W e remo ve bin types with small demand s and adjust the LP formulatio n an d the round in g s c heme u sed b y [5]. The run ning-time of the APT AS can then b e fur ther improv ed using the in volv ed metho d of Jansen and S olis-Oba [11] to yield the claimed AFPT AS in Theorem 26. Related W ork As already ment ioned, to the b est of our kno wledge, all of the previous w ork consid ers the (V ariable-Sized) Bin Covering p roblem in the in finite supp ly mo d el. Surveys on offline and online v ersions of these problems are giv en b y Cs ir ik and F renk [4] and by C s irik and W o eginger [7]. Historically , r esearc h (on the offline v ersion) of the Bin Co vering problem was initiated by Assmann et al. [1]. Th ey p ro ved that Next Fit is a 2-appro ximation algorithm. F urthermore, they prov ed that Firs t Fit Decreasing is an asymptotic 3  2-appro ximation and ev en impro v ed on this result b y giving an asymptotic 4  3- appro x im ate algorithm. Csir ik et al. [3 ] also obtained asymptotic appro xim ation guaran tees of 3  2 and 4  3 with simpler h eu ristic algorithms. The next breakthrough was made by Csirik, John son, and Keny on [5] by giving an AP- T AS for th e classical Bin Covering problem. Th e algorithm is based on a suitable L P relaxation and a r ounding sc h eme. Later on, Jan s en and Solis-Oba [11] impro ved up on the runn in g time and ga v e an AFPT AS. They reduce the num b er of v ariables by approxima ting the LP f ormulation of Csirik et al. [5], w hic h y ields the desired sp eed-up . Csirik and T otik [6] ga v e a lo w er b oun d of 2 for ev ery online algorithm for online (V ariable-Sized) Bin Co v - ering , i. e., items arriv e one-by-o n e. This b ound holds also asym p totically . Moreo ver, for V ariable-Sized Bin Covering any online algorithm must hav e an u n b ound ed appro xima- tion guarantee , wh ich can b e seen fr om th e f ollo wing easy construction: There are tw o bins with d emands d 1 = n and d 2 = 1. The first arriving item h as size 1. If an online algorithm assigns this to bin 1, no furth er items arr iv e. Otherwise, i. e., the item is assigned to bin 2, an item of size n − 1 arriv es. This item can also only b e assigned to bin 2 and h en ce an algorithm gains p rofit at m ost 2 d 2 = 2. It follo ws th e comp etitiv e ratio is at the b est equal to n  2. Th u s this online mo del is only in teresting from an asymptotic p ersp ectiv e. F or on lin e V ariable-Sized Bin Co ve ring it is easy to see that the algorithm Next Fit whic h uses only the largest bin type is already an asymptotic 2-a p proxima tion. T his in com bin ation with the b ound of C sirik and T otik [6] already s ettles the online case. B y con trast, there could b e reasonable appro ximation guaran tees in the offline mo del, b oth, n on-asymptotically and with unit supp ly . Ho we v er, to the b est of our kn o wledge, no non-asymp totic offline v ersion of V ariable-Sized or General ized Bin Covering h as b een considered p reviously . 4 Notation F or an y set K ⊆ J d efine th e total size by s ( K ) = ∑ k ∈ K s k . Note that a bin i ∈ I is co v ered b y a set K ⊆ J , if s ( K ) ≥ d i . As a shorthand , defi n e s = s ( J ) . Any assignment of items to b ins is a solution of the Gener alized Bin Covering pr ob lem. W e will denote suc h an assignment by a collectio n of sets S = ( S i ) i ∈ I , w here the S i ⊆ J are p airwise disjoin t subsets of the set J of items. Denote th e profit of a solution S by p ( S ) = ∑ i ∈ I ∶ s ( S i )≥ d i p i . The profit of a solution S determined by some algorithm alg on an ins tance ( I , J ) is denoted b y alg ( I , J ) = p ( S ) . W e may omit th e in stance ( I , J ) in calculations, if it is clear to wh ich instance alg refers to. F urthermore, for a solution S of an algorithm alg , let u alg ( i ) = s ( S i ) b e the total s ize of the items assigned to bin i . If no confusion arises, we will w rite u ( i ) instead of u alg ( i ) . 2 Generalized Bin Covering Theorem 1. Ther e exists a 5 -appr oximation f or General ized Bin Covering in the unit supply mo del, which has running time O ( nm √ m + n ) . In terms of lo wer b oun ds, recall that the problem is inap p ro ximab le u p to a factor of tw o, unless P = NP . In terms of u pp er b ounds it is n ot h ard to see that naive greedy strategies as that assign items to most profitable bins or that assign items to bins with the b est ratio of profi t to demand do not yield a constan t app ro ximation ratio. W e firs tly giv e an informal description of the ideas of our algorithm and defin e terms b elo w . A t the heart of our analysis f or the upp er b ound lies the follo win g observ ation. An optimal algorithm either co v ers a n ot to o small fr action of bin s with only one item exceeding th e demand of the r esp ectiv e bin or a large fraction of bins is co v ered with more than one item, and all these items are smaller than th e demand of the b in they were assigned to. W e explain b elo w w hy this can assumed to b e tru e. In the former case we sp eak of s in gular co v erage and in the latter of regular co v erage. It is easy to see (cf. Ob serv ation 2 ) that a b ipartite maxim um matc h ing giv es a solution b eing at least as go o d as the partial optimal solution of singularly co v ered bins. The more difficult case we h a ve to handle is when a large fraction of b ins is co v ered regularly in an optimal solution. W e manage this problem by considering an appropr iate mo d ified Bin Covering p roblem. In this p roblem items are only allo wed to b e assigned to bins with d emand of at most their size. In this s itu ation w e sa y that the items are admiss ible to the resp ectiv e b ins. F u rther we are allo wed to sp lit items in to parts and th ese parts may b e distribu ted among the bins to which the whole item is admissible. In tuitiv ely , in this mo dified problem th e pr ofit gained f or a bin is the f raction of demand co v ered m ultiplied with the profit of th e r esp ectiv e bin. In Lemma 3 we show that the mo dified problem can b e solved optimally in p olynomial time b y algorithm alg ∗ defined in Figure 1. Algorithm alg ∗ considers bins in non-increasing order of efficiency , where the efficiency of a bin is defin ed as the ratio of profit to d emand of the resp ectiv e b in. F or eac h b in i alg ∗ considers the largest item j , wh ic h is admissible to i . If j w as not assigned or only a part of j was assigned p reviously then j r esp ectiv ely the remaining part of j is assigned to i . T hen alg ∗ pro ceeds with the next smaller item. Once a bin is co vered, the item whic h exceeds the bin is split so that the bin is exactly co v ered. Note that it can h ap p en that durin g this pro cedur e bins are assigned items, but are not co v ered. But due to the defin ition of the mo dified p roblem, these bin s p rop ortionally con tribu te to 5 the ob jectiv e fu n ction. A solution found by this algorithm is optimal, w hic h we show b y transforming an optimal solution to a lin ear p rogram formulation of this mo difi ed pr oblem in to the solution of alg ∗ without losing any pr ofit. A s olution of alg ∗ can b e tr an s formed v ia t wo steps into a go o d solution for the Gene r- alized Bin Covering problem. By th e w a y alg ∗ splits items we are able to reassem b le the split items in Lemma 5 without losing to o muc h profit in the mo difi ed mo del. The solution is further mo dified in a greedy wa y suc h that there are no items on a not cov ered b in i , whic h are admissible to an other not co v ered bin i ′ with larger efficiency . A solution with this prop erty is called maximal w ith resp ect to the mo d ified Bin Co vering problem. F rom a m aximal solution we can create a solution for the Generalize d Bin Covering problem in Lemma 6, again by losing only a b ounded amount of profit. F or this we mov e items successiv ely from a n ot cov ered b in to the next not co v ered b in, wh ic h h as at least the same efficiency . Since the solution was maximal the bin s with higher efficiency are co vered. By this pro cedure all bins are co v ered, whic h w ere n ot co vered in the maximal solution, except the least efficien t one. Either this least efficien t bin or the remaining ones hav e at least half of the pr ofit of all bin s, w hic h w ere not co vered in the maximal solution. T h erefore, after applying this pro cedu re at most h alf of the p rofit is lost in comparison to the maximal solution. But n o w, all bins that receiv e items after this pro cedure are actually cov ered. W e start with th e definitions we need in order to pro ve of Theorem 1. Let S = ( S 1 , . . . , S m ) b e a solution. During the analysis we can assume that there are no unassigned items, i. e. all considered algorithms can assign all items, w hic h is form ally S 1 ∪ ⋅ ⋅ ⋅ ∪ S m = J . This is justifi ed since we could add a dummy bin m + 1 with p m + 1 = 0 and d m + 1 = ∞ for sak e of analysis. A co v ered bin i is called to b e co v ered singularly if S i = { j } for some j ∈ J with s j > d i , otherwise it is called to b e co vered r e gu larly . Since we can assume that all items can b e assigned to bin s, we can also mak e the follo w in g assumptions. A bin i which contai ns an item j with s j > d i is singularly co vered. F or a bin i w hic h is co vered r egularly it holds s ( S i ) ≤ 2 d i . The latter can b e assumed to b e tru e, since the b in i d o es n ot con tain an item j with s j > d i and hence in case s ( S i ) > 2 d i w e could remo v e an item and the b in i still would b e co vered. Observ ation 2. L e t ( I , J ) b e an instanc e and fix an optimal solution O on this instanc e. L et I S the bi ns c over e d singularly in O and J S the set of i tems on the bins I S . Ther e is an algorithm alg such that alg ( I , J ) ≥ opt ( I S , J S ) . The running time is O ( nm √ m + n ) . Pro of. W e define an alg orithm alg , w h ic h simply s olves the follo wing instance for Maximum Weight Bip ar tite Ma tching optimally: Define a bipartite graph G = ( I ∪ J, E ) with edge s E = { ij  s j > d i } and a w eight f unction w ∶ E → R giv en b y w ij = p i for ij ∈ E . Our algorithm alg determines a Maximum Weight Bip ar tite Ma tch ing M ⊆ E . Since our graph has m + n no des and at most mn edges th e algorithm of Hop croft and Karp [10] giv es a solution in time O ( nm √ m + n ) . The ind uced solution S = ( S 1 , . . . , S m ) is S i = { j } if ij ∈ M and S i = ∅ otherwise. Clearly , the s ingularly co ve red bins I S and the items J S assigned to them corresp ond to a matc hin g in G . Thus alg ( I , J ) ≥ alg ( I S , J S ) = opt ( I S , J S ) by the correctness of the matc hing algo rithm. 6 Consider the follo wing mo d ified Bin Co v ering pr oblem. I tems may b e sp lit int o p ≥ 1 parts. Then w e consid er an item j as p items ( j, 1 ) , . . . ( j, p ) of p ositiv e s ize, where w e m ay omit the b races in indices. Denote s j,i the s ize of item part ( j, i ) of item j . F ormally it has to hold s j = ∑ p i = 1 s j,i and s j,l > 0 for 1 ≤ l ≤ p . W e refer to the ( j, l ) as the parts of the item j . An item j is said to b e admissible to a bin i , if s j ≤ d i . The parts ( j, l ) of an item j are defined to b e admissib le to i if and only if j is admissible to i . Item parts can only b e assigned to bins to w h ic h th ey are admissible. F or a fi x ed solution S = ( S 1 , . . . , S m ) let S i b e the set of item parts, assigned to b in i . Let y i ∶ = min { s ( S i ) d i , 1 } b e the fill lev el of b in i (note th at the fill level of bin i m a y b e at most one, but nev ertheless s ( S i ) > d i is p ermitted, i. e. the su m of item sizes assigned to b in i may exceed its demand). The profit gained f or b in i in th e mo dified problem is p ∗ ( S i ) ∶ = p i y i , whic h is intuiti v ely the p er centag e of co v er ed demand multiplied with the profit of the bin , where the maximal profit whic h can b e gained is b ounded by p i . F u r ther for a set I ′ ⊆ I of bins and a solution S = ( S 1 , . . . , S m ) let p ∗ ( I ′ ) = ∑ i ∈ I ′ p ∗ ( S i ) . W e define the efficiency e i of bin i to b e e i ∶ = p i  d i . W e define th e algorithm alg ∗ in Figure 1 f or the mo dified Bin Covering problem and denote as usual w ith alg ∗ ( I , J ) the v alue of its solution for the mo d ified problem on the instance ( I , J ) . Analo gously denote opt ∗ ( I , J ) the v alue of an optimal solution to the mo dified Bin Covering pr oblem. • Sort b ins n on-increasingly by efficiency an d assume e 1 ≥ ⋅ ⋅ ⋅ ≥ e m . • Set x j,i ∶ = 0 for j = 1 , . . . , n and i ∶ = 1 , . . . , m . • Let x j = m ∑ i = 1 x j,i . • F or i = 1 , . . . , m do While bin i is not co v ered do – If ∄ j ∈ J ∶ s j ≤ d i ∧ x j < s j , then stop execution of this while lo op, and pr o ceed with the next bin i + 1. – Other w ise c ho ose a largest item j with x j < s j and s j ≤ d i . – If ∑ n l = 1 x l,i + ( s j − x j ) ≤ d i then x j,i ∶ = s j − x j . [assign the remaining p art of item j .] – Else x j,i ∶ = d i − ∑ n l = 1 x l,i . [assign only a part, su c h that i is co vered] • Ou tput th e solution S induced by the x j,i v ariables. [cf. text b elo w LP (1)] Figure 1: Th e algorithm a lg ∗ . Our mo difi ed Bin Covering problem can b e form u lated as a linear program (LP), whic h will simp lify th e descrip tion of the analysis of th e up coming algorithm for the mo dified problem. Note that we do not need to solv e this linear p rogram. Actual ly , a l g ∗ solv es it 7 optimally . Moreo v er, Lemma 3 and Lemma 5 imply a in tegralit y gap of t wo for this linear program and the corresp onding intege r linear p rogram. maximize m  i = 1 p i y i (1) sub ject to y i ≤ n  j = 1 x j,i  d i ∀ i ∈ I m  i = 1 x j,i ≤ s j ∀ j ∈ J 0 ≤ y i ≤ 1 ∀ i ∈ I 0 ≤ x j,i ∀ i ∈ I ∀ j ∈ J 0 ≥ x j,i ∀ i ∈ I ∀ j ∈ J with s j > d i W e may iden tify th e x j,i v alues with the s izes s j,l , where x j,i = 0 means in fact th at no part of item j was assigned to bin i . If there are p v alues x j,l > 0 for a fixed j then this means that alg ∗ splits the item j in to p parts, and ther e are p v alues s j, 1 , . . . , s j,p > 0. Lemma 3. Algor ithm alg ∗ gives a solution of v alue alg ∗ ( I , J ) = opt ∗ ( I , J ) . Pro of. W e show that our algorithm giv es an optimal solution to LP (1) by transforming an arbitrary optimal solution to LP (1 ) in to a solution foun d b y our algorithm without losing an y profit. Let O = ( O 1 , . . . , O m ) b e an optimal solution and assume ( y ′ i , x ′ j,i ) 1 ≤ i ≤ m, 1 ≤ j ≤ n are the corresp onding v ariables d escribing the assignmen t of the item p arts to bins in O . Let S b e the solution found b y alg ∗ and let ( y i , x i,j ) 1 ≤ i ≤ m, 1 ≤ j ≤ n b e the corresp ondin g v ariables set b y alg ∗ . In iteration i = 1 , . . . , m w e set y ′ i ∶ = y i and x ′ j,i ∶ = x j,i for all j ∈ J and s ho w that the optimalit y is preserved. Since items are arbitrary splittable and th e assum p tion that all items are assigned by a lg ∗ and opt we can assume that s ( S i ) , s ( O i ) ≤ d i . Consider bin i of the optimal solution and assume the bins 1 , . . . , i − 1 in th e optimal solution con tain only the items, whic h are assigned to these bin s by the solution of alg ∗ , i. e. we hav e already x ′ j,l = x j,l for all 1 ≤ l ≤ i − 1 and all j ∈ J . Case s ( S i ) < d i : This means that all items b eing admissible to bin i were assigned to bin i or to bins with ind ices 1 , . . . , i − 1 in the solution S by construction of the algorithm alg ∗ . F orm ally we h a ve thus x j,i ′ = 0 for all j ∈ J with s j ≤ d i and all i ′ > i . S ince for all v ariables x ′ j,i ′ = x j,i ′ for all i ′ < i and j ∈ J holds, it follo w s th at ∑ n j = 1 x ′ j,i ≤ ∑ n j = 1 x j,i . If ev en ∑ n j = 1 x ′ j,i < ∑ n j = 1 x j,i then the “missing” item parts m u st b e assigned to bins i ′ > i in th e optimal solution O . And so, if x ′ j,i < x j,i for some j , w e set x ′ j,i ∶ = x j,i and x ′ j,i ′ ∶ = 0 for all i ′ > i . W e increase the y i v ariable and decrease the y i ′ for i ′ > i corresp onding to the c h an ges. By this pro cess the ob jectiv e v alue can only b e incr eased, since e i ≥ e i ′ for all i ′ > i and the sum of y i v ariables maint ains the same. Also note that no constrain ts are violated, since a lg ∗ assigns items only to bins, to which they are admissib le. Case s ( S i ) = d i : If s ( O i ) < d i then again the “missing” items ha ve to reside on bins i ′ > i and we can increase some x j,i v ariables as ab o v e su c h th at ∑ n j = 1 x ′ j,i = ∑ n j = 1 x j,i holds. No w , if x ′ j,i ≠ x j,i for some j ∈ J then there hav e to b e items j, j ′ ∈ J s u c h that w e hav e for the 8 corresp ondin g v ariables x j,i > x ′ j,i and x j ′ ,i < x ′ j ′ ,i . F or the transformation w e pro cee d in the ordering as al g ∗ did: Let j 1 , . . . , j k the items assigned to bin i by al g ∗ in this ord ering, i. e. alg ∗ c hanged the v alues of the v ariables x j 1 ,i , . . . , x j k ,i in this order in g. Then we fin d the item j l ∈ { j 1 , . . . , j k } with smallest index 1 ≤ l ≤ k such that x j l ,i > x ′ j l ,i holds and an arbitrary item j ′ suc h that x j ′ ,i < x ′ j ′ ,i holds. F or ease of n otation w e set j ∶ = j l . Let i ′ > i b e the index of a bin suc h that x ′ j,i ′ > 0. Clearly , su c h an ind ex m u st exist by our assumption x ′ j = x j = s j holds and since x ′ j,i ′ = x j,i ′ for all i ′ < i and all j ∈ J . As x j,i > 0 and x ′ j,i ′ > 0 it follo ws that item j is admissible to i and i ′ . Assume we already hav e sho w n s j ≥ s j ′ . Note, this is not a pr iori clear, since al g ∗ ma y split items arbitrarily . Then, as x ′ j,i ′ > 0 item j is admissible to bin i ′ and so is the item j ′ , b ecause of s j ≥ s j ′ . Let δ ∶ = min { x ′ j,i ′ , x ′ j ′ ,i } . W e set x ′ j,i ∶ = x ′ j,i + δ , x ′ j,i ′ ∶ = x ′ j,i ′ − δ , x ′ j ′ ,i ∶ = x ′ j ′ ,i − δ and x ′ j ′ ,i ′ ∶ = x ′ j ′ ,i ′ + δ . This pro cess of adju sting can b e iterated until x j,i = x ′ j,i for all j ∈ J . W e are left to show s j ≥ s j ′ . W e argue this comes ind eed from the fact that the algorithm alg ∗ considers items in non-in cr easing order of size. Assume to the contrary that s j ′ > s j . Then alg ∗ w ould hav e consid ered item j ′ b efore item j . Because of x ′ j ′ ,i > 0 it must b e in fact the case th at al g ∗ assigns a part of item j ′ to b in i otherwise we had x ′ j ′ ,i = 0 b y the prop erty x j,i ′ = x ′ j,i ′ for all i ′ < i and all j ∈ J . Then either x j ′ ,i ∶ = s j ′ − x j ′ w as set (in case ∑ n l = 1 x l,i + ( s j − x j ) ≤ d i ), whic h in tuitiv ely means that the y et unassigned rest of the item j ′ w as assigned to bin i b y alg ∗ . In this case x j ′ ,i ′ = 0 for all i ′ > i and hen ce it follo ws x ′ j ′ ,i ≤ x j ′ ,i , again b ecause of the prop ert y x ′ j ′ ,i ′ = x j ′ ,i ′ for all i ′ < i . T h is is a con tradiction to the assumption x ′ j ′ ,i > x j ′ ,i . Hence it m u st hav e b een the case that x j,i ∶ = d i − ∑ n l = 1 x l,i w as set b y th e algorithm. W e had x j ′ + ∑ n l = 1 x l,i > d i in this situation, i. e. n ot the whole remaining size of item j ′ w as assigned to bin i . Th en w e conclude j ′ w as the last item assigned to bin i by alg ∗ . This con tradicts the f act th at item j is assigned to bin i by alg ∗ , since we ha ve assumed that x j,i > x ′ j,i ≥ 0. This concludes the p ro of of the lemma. Observ ation 4. L et S b e a solution with the pr op erty s ( S i ) ≤ d i for al l i ∈ I . L et S ∗ b e a solution, with the pr op erty s ( S ∗ i ) ≤ 2 d i for al l i ∈ I . If for al l item p arts j ∈ J ther e holds j ∈ S i and j ∈ S ∗ i ′ , wher e e i ≤ e i ′ , then p ∗ ( S ) ≤ 2 p ∗ ( S ∗ ) . Pro of. If s ( S i ) ≤ d i for all i ∈ I then we can compute p ∗ ( S ) “itemwise” p ∗ ( S ) =  i ∈ I s ( S i ) e i =  i ∈ I  j ∈ S i s j e i ≤  i ∈ I  j ∈ S ∗ i s j e i =  i ∈ I s ( S ∗ i ) e i (2) ≤  i ∈ I 2 min { s ( S ∗ i ) d i , 1 } d i e i (3) = 2  i ∈ I p ∗ ( S ∗ i ) = 2 p ∗ ( S ∗ ) , where In equalit y (2) is by e i ≤ e i ′ for i > i ′ and th e fact that we assign items only to bin s with s maller ind ices in S ∗ in comparison to solution S and In equ alit y (3) holds, since b y 9 precondition we ha v e for eac h i ∈ I that ∑ j ∈ S ∗ i s j  2 ≤ d i in S ∗ . Hence the claim follo w s. W e call a solution S maximal with resp ect to the mo dified Bin Covering p roblem, if there are no tw o distinct bins i and i ′ with 0 < s ( S i ) < d i and 0 < s ( S i ′ ) < d i ′ and e i ≤ e i ′ , suc h that there is an item j ∈ S i , w hic h is ad m issible to bin i ′ . Note that th is implies the follo wing for suc h bins i and i ′ . If we assign in a maximal solution only one item from a b in i to a bin i ′ then bin i ′ is already co vered by only th is item. W e sa y a solution S cont ains no split items, if for all i ∈ I and j ∈ S i w e ha ve s j, 1 = s j . Lemma 5. L et S b e a solution give n by al g ∗ for the mo difie d pr oblem. Then ther e exists a solution S ∗ such that p ∗ ( S ) ≤ 2 p ∗ ( S ∗ ) and S ∗ c ontains no split items. F urther S ∗ is maximal with r esp e ct to the mo difie d Bin Covering pr oblem. Pro of. In a fir st step w e crea te a solutio n S ′ suc h that S ′ con tains n o split items, i. e. s j, 1 = s j for all j ∈ J . Let j b e an item, whic h is sp lit by alg ∗ in to p > 1 parts ( j, 1 ) , . . . ( j, p ) . W e “merge” th e p arts ( j, 1 ) , . . . , ( j, p ) the follo win g wa y . W e assign all the parts ( j, 2 ) , . . . , ( j, p ) to the bin , to whic h ( j, 1 ) was assigned. T he solution created is the solution S ′ . W e argue that b y th is pro cedure eac h b in receiv es p arts of at most one item. By the algorithm, an item j is split only then, when it fills a b in. The bin filled is assigned a part ( j, l ) and neve r r eceiv es an item (part) after that. Hence eac h b in i receiv es at most one first part of an item, i. e. if ( j ∗ , 1 ) ∈ S i then ( j, 1 ) ∉ S i for j ≠ j ∗ . It follo ws that eac h b in is assigned the parts of only one item, since parts ( j, l ) are alwa ys assigned to the bin i with ( j, 1 ) ∈ S i . Assume item j was split in to p > 1 parts ( j, 1 ) , . . . , ( j, p ) by alg ∗ . If a part ( j, 1 ) was assigned to a b in i then part ( j, 1 ) wa s admissible and so w as item j . Thus s j ≤ d i . It follo ws that s j, 1 + ⋅ ⋅ ⋅ + s j,p ≤ d i . As w ith the ab ov e argumentatio n eac h bin i receiv es only parts of one item j and we had s ( S i ) ≤ d i , b y the algorithm al g ∗ , it follo w s s ( S ′ i ) ≤ s ( S i ) − s j, 1 + s j, 1 + s j, 2 + ⋅ ⋅ ⋅ + s j,p < 2 d i , b ecause s j, 1 > 0. Hence we ha ve sh o wn that s ( S ′ i ) < 2 d i for eac h bin i ∈ I in the solution S ′ . As already describ ed, w hen al g ∗ splits an item j into p > 1 p arts, then the fir st p art ( j, 1 ) fills a bin. Since alg ∗ considers the items in n on-increasing ord er of efficiency , w e h a ve that the parts ( j, 2 ) , . . . , ( j, p ) are assigned to bins with at most the same efficiency . Hence in the solution S ′ eac h item is assigned to a bin i with at least the same efficiency as the bin i ′ , to whic h it w as assigned in the solution S . The solution S ′ can no w b e transf ormed in to th e maximal solution S ∗ preserving the b oth men tioned prop er ties, whic h are the p reconditions of Observ ation 4 . F or this let T ∶ = { i ∈ I  0 < s ( S ′ i ) < d i } . Let T = { e i 1 , . . . , e i l } and assu m e as usual e i 1 ≥ ⋅ ⋅ ⋅ ≥ e i l . F or j = 1 , . . . , l do the follo win g. While bin i j is not co v ered and one of the bins i j + 1 , . . . , i l con tains an item j ′ , which is admissib le to i , assign j ′ to bin i j . If i j is co ve r ed or there are no items left on the bins i j + 1 , . . . , i l , which are admissible to i j , then p ro ceed with bin i j + 1 and so on. T he so created solution is th e s olution S ∗ . Clearly , items are only assigned to more efficien t bin s. F u rther, an item j is only assigned to a bin i , when j is adm iss ible to i and i is not y et co vered. Hence, it follo ws s ( S ∗ i ) ≤ 2 d i . Moreo v er, if s ( S ∗ i ) < d i for a bin i , then by construction all bins i ′ ∈ T with e ′ i ≤ e i do not con tain an item j ′ whic h is admissible anymore. Hence S ∗ is also maximal w. r. t. the mo d - ified Bin Covering problem. Applying Observ ation 4 to the solutions S and S ∗ giv es the 10 claim of the lemma. Lemma 6. L et S b e a solution b eing maximal with r esp e ct to the mo difie d Bin Covering pr oblem and c ontaining no split items. Then ther e exist a solution S ∗ for the Gener alized Bin Co vering pr oblem, such that p ∗ ( S ) ≤ 2 p ( S ∗ ) . Pro of. L et R ∶ = { i ∈ I  0 < s ( S i ) < d i } . Assume w . l. o. g. that R = { 1 , . . . , l } and e 1 ≥ ⋅ ⋅ ⋅ ≥ e l . Construct tw o solutions S ′ and S ′′ . W e set S ′ l = J and S ′ i = ∅ f or 1 ≤ i ≤ m, i ≠ l . F urther we set S ′′ i − 1 = S i for 2 ≤ i ≤ l , S ′′ l = ∅ and S ′′ i ∶ = S i for l < i ≤ m . In S ′ the only bin l whic h is assigned items is cov ered , since we ma y assume w. l. o. g. that eac h bin i is co v ered , when all items are assigned to it. Th e same holds true for S ′′ : Th e bins from { l + 1 , . . . , m } , whic h con tain items, are co ve red sin ce th ey were already co ve r ed in S . The b in s 1 , . . . , l − 1 are co v ered, s ince S is a maximal solution with r esp ect to the mo dified Bin Co vering problem and e i ≥ e i + 1 holds f or 1 ≤ i ≤ l − 1. Finally we hav e S ′′ l = ∅ . As n one of the solutions S ′ and S ′′ con tains split items and all b ins are co vered, we hav e that p ∗ ( S ′ ) = p ( S ′ ) and p ∗ ( S ′′ ) = p ( S ′′ ) . W e output S ∗ ∶ = S ′ if p ( S ′ ) = max { p ( S ′ ) , p ( S ′′ )} and S ∗ ∶ = S ′′ otherwise. T o see the claim ab out the appro ximation guaran tee distinguish the cases p l > p ∗ ( S ) 2 and p l ≤ p ∗ ( S ) 2, where the index l is as ab o v e th e in d ex of the bin with S ′ l = J . If p l > p ∗ ( S ) 2 then p ( S ∗ ) ≥ p ( S ′ ) = p l > p ∗ ( S ) 2. If p l ≤ p ∗ ( S ) 2 then p ( S ∗ ) ≥ p ( S ′′ ) = p ∗ ( S ) − p l ≥ p ∗ ( S ) − p ∗ ( S ) 2 = p ∗ ( S ) 2, whic h concludes the pro of of the lemma. Pro of (of Theorem 1). L et ( I , J ) b e the giv en instance. O ur algorithm wo rks as follo w s. W e use Ob serv ation 2 to fi nd a solution S 1 . Then we ru n alg ∗ on the ins tance ( I , J ) and let S b e the solution outpu t. W e transform the S into a solution S ′ as done in Lemma 5 and then solution S ′ in to solution S 2 as done in Lemma 6 . W e outpu t the b etter solution from { S 1 , S 2 } . The run ning time is clearly dominated by th e algorithm for M aximum We ight Bip ar tite Ma tching . W e give the pr o of on the appro ximation guaran tee. Fix an optimal solution O to the instance ( I , J ) . Let I R ⊆ I b e the set of b ins co vered regularly by the solution O and J R = { j ∈ J  ∃ i ∈ I R ∶ j ∈ O i } , the set of items on these bins. Let I S ⊆ I b e the set of bins cov ered singularly by the solution O and J S = { j ∈ J  ∃ i ∈ I S ∶ j ∈ O i } , the set of items on these bins. W e hav e opt ( I , J ) = opt ( I R , J R ) + op t ( I S , J S ) . Th us opt ( I , J ) − opt ( I R , J R ) = op t ( I S , J S ) . Case opt ( I R , J R ) < 4  5 ⋅ opt ( I , J ) . T hen op t ( I S , J S ) > 1  5 ⋅ opt ( I , J ) by the ab o ve. Hence in this case we outp ut a solution such that op t ( I , J ) ≤ 5 alg ( I , J ) b y O bserv ation 2 . Case opt ( I R , J R ) ≥ 4  5 ⋅ opt ( I , J ) . It follo ws opt ( I S , J S ) ≤ 1  5 ⋅ opt ( I , J ) . W e fin d opt ( I , J ) = opt ( I R , J R ) + opt ( I S , J S ) ≤ opt ∗ ( I R , J R ) + 1  5 ⋅ o pt ( I , J ) (4) ≤ opt ∗ ( I , J ) + 1  5 ⋅ op t ( I , J ) = alg ∗ ( I , J ) + 1  5 ⋅ opt ( I , J ) (5) ≤ 4 alg ( I , J ) + 1  5 ⋅ opt ( I , J ) . (6) In In equalit y (4) w e use opt ∗ ( I R , J R ) ≥ opt ( I R , J R ) and th e assump tion of the case. In Inequ alit y (5) w e use Lemma 3. In Inequalit y (6) we ha ve accoun ted for transf orming 11 the fractional s olution to the mo dified problem into a solution for the Generalize d Bin Co vering problem with Lemm as 5 and 6. It follo ws op t ( I , J ) ≤ 5 alg ( I , J ) . 3 V ariable-Sized Bin-Cover ing 3.1 A Tight Analysis of Next Fit Decreasing in the Unit Supply Mo del In this subsection, we h a ve d i = p i for all i in th e unit supp ly mo del. The algorithm Next Fit Decreasing ( nfd ) is giv en in Figure 2. The algorithm considers bin s in non-increasing order of demand. F or eac h b in, if th e total size of the unassigned ite ms suffices for co verage , it assigns as many items (also non-in cr easing in size) as n ecessary to co ver th e bin. Otherwise, the b in is skipp ed. In this section w e assume that we ha v e d 1 ≥ ⋅ ⋅ ⋅ ≥ d m and s 1 ≥ ⋅ ⋅ ⋅ ≥ s n , as needed by the algorithm. • Sort b ins n on-increasingly by d emand and rename the bins such that d 1 ≥ ⋅ ⋅ ⋅ ≥ d m . • Sort items non-increasingly by size and r en ame the items such that s 1 ≥ ⋅ ⋅ ⋅ ≥ s n . • Let i = 1 b e the curr ent bin and j = 1 th e in d ex of the firs t u nassigned item. • While j ≤ n and i ≤ m d o – If ∑ n l = j s l < d i set i ∶ = i + 1 and S i = ∅ . – Else let j ′ b e the s mallest index with ∑ j ′ l = j s l ≥ d i . Assign the items j, . . . , j ′ to bin i , i. e., S i = { j, . . . , j ′ } Set i ∶ = i + 1 and j = j ′ + 1. • Return S = ( S i ) i ∈ I . Figure 2: Algorithm nf d . Example 7. L et 2  3 > ǫ > 0 b e arbitr ary. The fol lowing instanc e ( I , J ) yi e lds that nfd gives an appr oximation not b etter than 9  4 − 2 ǫ . H enc e nf d is at le ast a 9  4 -appr oximation. L et I = { 4 , 3 − 2 ǫ, 3 − 2 ǫ, 3 − 2 ǫ } and J = { 2 − ǫ, 2 − ǫ , 2 − ǫ, 1 − ǫ, 1 − ǫ, 1 − ǫ } . O b serve we have nfd ( I , J ) = 4 and op t ( I , J ) = 9 − 6 ǫ . Theorem 8. nf d is a 9/4-appr oximation algorithm with running time O ( n log n + m log m ) . The b ound is tight. Note that this is almost b est p ossible, since the problem is inappro ximable u p to a factor of t wo , unless P = NP , whic h holds for unit sup ply ev en asymp totically in the notion of Theorem 25. Pro of tec hniques. W e w ill use three kinds of arguments. Th e fi rst t yp e w e call a vol ume argumen t. If s is th e sum of item sizes in the (remaining) instance, we ha ve opt ≤ s . This argumen t h olds indep end en tly of the concrete demands of bin s. S uc h vol ume b ounds are to o w eak in general to achiev e the claimed b ound, th u s we need arguments using the s tructure 12 of bins in the instance, w h ic h is the second t yp e of argumen ts. F or example, if the sum of item sizes in the (remaining) instance is αd , α > 1 and the d emand of the only bin in the instance is d , then it follo ws opt ≤ d , while we could only conclud e opt ≤ αd with a v olume argumen t. Th e third typ e of argumen t we u se are arguments transf orming instances. These argumen ts giv e that we can w. l. o. g. restrict our selv es to analyze instances having certain prop erties. F or example, we may assume that ther e are no items in the in stance with size larger than the largest b in demand. Pro of outline. Our pr o of lo oks at the sp ecific stru cture of the s olution giv en by n fd and argues b ased on that, how m uc h b etter an optimal solution can b e. W e emplo y the d escrib ed tec hniques in th e follo win g w ay . Firstly , we settle t wo basic prop erties of nfd : A solution of nfd is unique (Ob s erv ation 9) and if a bin is co v ered w ith at least t w ice its demand, then there is only one item assigned to it (Observ ation 10). These p rop erties will b e used implicitly dur ing the an alysis. After that, w e give tr ansformation arguments, which allo w us to restrict ours elv es to analyze in s tances with the follo wing prop erties. W e may assume that n fd co v ers the first bin (Observ ation 11 ), and that the “righ t-most bins ” (i. e. the bins with the least d emand – or the smallest bins) are empt y (Observ ation 12), where we will sp ecify this notion in more detail later. W e will sho w that we m a y assu me that the “left-most bins” (i. e. the largest b ins) are only assigned items su c h th at they do n ot exceed t wice their demand (Lemma 13). Here “left-most bins” refers to th e bins up to the fi rst empt y bin. With these to ols at hand we can come to the actual p ro of. Th e central n otion h er e is the wel l-c over e d bin (Definition 14): Consider the r igh t-most (i.e., smallest) empty bin in the in stance with the prop ert y that all larger b in s are assigned items only up to t wice their demand. If suc h a b in exists, then we call th e co vered b ins of these well-c o ve r ed. T he pr o of will b e indu ctiv e. The terminating cases are the ones, w hen there are either at least four w ell-co vered bins (Ob s erv ation 16) or b et ween t wo and three wel l-co v ered bins, but there is a b in among these con taining at least three items (Lemma 21). These cases are settled by v olume arguments which is th e r eason, why they are termin ating cases – ev en if there are additional filled b ut not w ell-co vered bins in the instance. W e are also in terminating cases if the ab o v e prerequisites are not met, b u t there are no filled bins w hic h are not we ll-co vered: Lemma 17 treats the case that all of the at m ost three w ell-co vered b ins con tain at most tw o items and Lemma 18 give s the cases, in which we ha ve exactly one w ell-co vered bin in the instance. If there are additional filled but not wel l-co v ered bins and we cannot apply v olume argu- men ts – as in the b oth last mentioned situations – , w e hav e to lo ok at the instance more closely . O ur idea is here to consider a sp ecific not w ell-co vered bin, whic h will b e called the he ad of the instanc e . W e will sub d ivide such an ins tance into tw o p arts, w h ic h is done b y the k ey lemma of the r ecursion step, the Decomp osition Lemma 23. Therein and in Lemma 20 w e sho w th at it is not adv an tageous to assign items, which nfd assigned to bins with larger demand than the demand of the head of the instance, to b ins with smaller demand th an the demand of the head of the instance. Th is allo ws us in com bin ation with some estimations to consider th e left part of the instance and the righ t p art separately . F or the le ft part Lemma 22 and Lemma 23 giv e that the appro ximation factor of nfd is at most 9  4 and the right part of the instance is a smaller instance and we may hen ce iterativ ely app ly th e argumenta tion. W e n ow start with the pro of and giv e four observ ations, wh ic h are easy but also rather 13 imp ortant and will b e u sed often implicitly dur in g the analysis. Th e fir st observ ation is immediate from nfd ’s b eha vior, and thus n o pro of is necessary . Observ ation 9. Fix an instanc e ( I , J ) . Then the solution of nfd (up to r e naming items of identic al size ) for this instanc e is uni q ue. F urther, if nfd did not assign the item j 0 to a bin, then nfd do es not assign the items j 0 + 1 , . . . , n to a bin either. In the follo w ing we similarly assum e w. l. o. g. that if for any b in i w e h a ve u opt ( i ) > 0 then u opt ( i ) ≥ d i and if S i is the set of items assigned to a b in i , th en u opt ( i ) − s j < d i for an y j ∈ S i , i. e. opt d o es not assign items, which are not needed to co ve r a bin . The next observ ation giv es that, if a bin in nfd ’s solution is assigned at least twice its d emand, there is only one item on it. Recall S i is the set of items nfd assigns to b in i . Observ ation 10. If s ( S i ) ≥ 2 d i then  S i  = 1 . Pro of. Ass ume  S i  ≥ 2. Immediate f rom the algorithm we ha ve that nfd uses a n ew bin , if the current b in is fu ll and nfd neve r puts an item on a full b in . Hence let there b e only one item j ∗ ab o ve the b ound ary of d i , i. e. if 1 , . . . , j ∗ are the items, whic h nfd assigns to bin i , then w e ha v e u ( i ) − s j ∗ < d i and u ( i ) ≥ d i . If u ( i ) ≥ 2 d i , then it f ollo ws that s j ∗ > d i . But for all items j ∈ S i ∖ { j ∗ } it holds s j < d i , b eca use this even h olds for the su m of all these items. As  S i  > 1 this is a con tradiction to the fact that nfd assigns items in non-increasing order. The next observ ation giv es that we ma y restrict our selv es to the analysis of su c h instances, where nfd co vers the fi r st bin. Observ ation 11. Fix an instanc e ( I , J ) . If u nfd ( 1 ) = ⋅ ⋅ ⋅ = u nfd ( i ) = 0 then u opt ( 1 ) = ⋅ ⋅ ⋅ = u opt ( i ) = 0 . L et I ′′ = I ∖ { 1 , . . . , i } . Then nfd ( I , J ) = nfd ( I ′′ , J ) and opt ( I , J ) = opt ( I ′′ , J ) . Pro of. If u nfd ( 1 ) = ⋅ ⋅ ⋅ = u nfd ( i ) = 0, then from nfd ’s b ehavio r we kno w that the su m of all item sizes in the instance d id not su ffi ce to fi ll bin i . Hence, by the ordering of bins, it d o es not suffice to fill the bins 1 , . . . , i − 1 either. Of course, the same h olds true for opt . By the argumen t giv en by the p revious ob s erv ation it is also justified to assume nfd ( I , J ) > 0. Since otherwise also opt ( I , J ) = 0 follo ws and nfd is optimal. This assu mption will alw a y s b e implicitly used and thus the quotien t opt ( I , J ) nfd ( I , J ) is alw ays defined. Alternativ ely w e could d efine opt ( I , J ) nfd ( I , J ) ∶ = 1, if nf d ( I , J ) = opt ( I , J ) = 0. F ur ther, we ma y alw ays assume that th ere exists an empt y bin, otherwise nf d is clearly optimal. W e ma y strengthen this observ ation suc h that it su ffices to compare instances of nfd to opt , wh ere the righ t-most bins are all empty , i. e. th er e is a non-emp t y b in i ′ , th e bin i ′ + 1 is emp t y and all bins w ith higher indices, if they exist, are also empty . Observ ation 12. L et a solution of nfd for an instanc e ( I , J ) b e given. L et i ∗ b e a bin with u ( i ∗ ) = 0 and for al l i > i ∗ we ha ve u ( i ) > 0 . Then opt ( I , J )  nfd ( I , J ) ≤ opt ( I ′ , J ) nfd ( I ′ , J ) , wher e I ′ = I ∖ { i ∗ + 1 , . . . , m } . Pro of. Since nfd did not fi ll bin i ∗ w e kn o w by nfd ’s b eha vior d i ∗ > ∑ m l = i ∗ + 1 d l . Hence on the one hand for the instance ( I ′ , J ) w e ha ve nfd ( I ′ , J ) = nfd ( I , J ) − ∑ m l = i ∗ + 1 d l b y nfd ’s 14 b ehavio r . As we ma y assume th at bin 1 is filled, we hav e on the other hand nf d ( I , J ) ≥ d 1 ≥ d i ∗ > ∑ m l = i ∗ + 1 d l and it follo ws nfd ( I , J ) > ∑ m l = i ∗ + 1 d l . F or opt we hav e opt ( I ′ , J ) ≥ opt ( I , J ) −  l > i ∗ , u ( l ) > 0 d l , since opt can p ossibly assign the items, w hic h p otentia lly r eside on the bins i ∗ + 1 , . . . , m in its solution to the instance ( I , J ) in the instance ( I ′ , J ) to other b in s. Altogether w e fin d opt ( I ′ , J ) nfd ( I ′ , J ) ≥ opt ( I , J ) − ∑ l > i ∗ , u ( l ) > 0 d l nfd ( I , J ) − ∑ m l = i ∗ + 1 d l ≥ opt ( I , J ) − ∑ m l = i ∗ + 1 d l nfd ( I , J ) − ∑ m l = i ∗ + 1 d l ≥ opt ( I , J ) nfd ( I , J ) , where the last in equalit y h olds, b ecause of o p t ( I , J ) ≥ nfd ( I , J ) > ∑ m l = i ∗ + 1 d l > 0. With this observ ation let u ( m ) = 0 from now on. The next lemma states that we can assume w. l. o. g. that all bin s i up to the bin with smallest index i ∗ , su c h that u ( i ∗ + 1 ) = 0, receiv e only items in su c h a w ay , th at u ( i ) ≤ 2 d i for i < i ∗ . Lemma 13. L et ( I , J ) b e an instanc e and c onsider a solution of nfd for it. L et i ∗ b e the smal lest index, such that i ∗ is a bin with u ( i ∗ ) > 0 and u ( i ∗ + 1 ) = 0 . L et i 1 , . . . , i k ∈ { 1 , . . . , i ∗ } b e the indic es with u ( i j ) ≥ 2 d i j for j = 1 , . . . , k and let j 1 , . . . , j k b e the items on these bins. Set I ′ = I ∖ { i 1 , . . . , i k } and J ′ = J ∖ { j 1 , . . . , j k } . Then opt ( I , J ) nf d ( I , J ) ≤ opt ( I ′ , J ′ ) nfd ( I ′ , J ′ ) . Pro of. Consider b in i 1 and item j 1 , where we ha ve by Observ ation 10 that on i 1 in f act resides only one item. W e argue that we can assum e, an optimal algorithm also assigns j 1 to i 1 . By the orderin g of items, for every bin i ≥ i 1 , if w e assign j 1 to i , w e hav e that s j 1 − d i ≥ s j 1 − d i 1 = u ( i 1 ) − d i 1 . Because of this and b ecause eve ry bin i ≥ i 1 is co ve r ed with only the item j 1 w e can assu m e an optimal algorithm would assign j 1 to a b in w ith an index at most i 1 . If an optimal algorithm decides to assign j 1 to a bin with index smaller than i 1 , th en it w ould not assign all of the items 1 , . . . , j 1 − 1 to the b in s 1 , . . . , i 1 − 1. This is b eca u se also n fd assigns the items 1 , . . . , j 1 − 1 to the b in s 1 , . . . , i 1 − 1 and these are already co ve red, but the bin i 1 w ould not, in this case. Hence at least one of th e items 1 , . . . , j 1 w ould b e assigned to a bin with ind ex at least i 1 , otherwise this assignment w ould not b e optimal. With the same argumen tation as ab o ve, for every suc h item j , if it w ould b e assigned to a bin i ≥ i 1 w e had s j − d i ≥ s j − d i 1 . Because eve r y su c h item j co ve rs ev ery bin i ≥ i 1 alone, we can thus assum e j is assigned to i 1 . But then, since s j − d i 1 ≥ s j 1 − d i 1 , we also can assu me j 1 is assigned by opt to i 1 . Hence, defining I ′′ = I ∖ { i 1 } and J ′′ = ∖ { j 1 } we h a ve opt ( I , J ) nfd ( I , J ) = d i 1 + o pt ( I ′′ , J ′′ ) d i 1 + nfd ( I ′′ , J ′′ ) ≤ opt ( I ′′ , J ′′ ) nfd ( I ′′ , J ′′ ) , 15 where the last inequalit y is b ecause opt ( I ′′ , J ′′ ) ≥ n fd ( I ′′ , J ′′ ) > 0. Iterativ ely applying this argumen tation yields the statemen t. In order to simp lify the analysis we defin e the n otion of a wel l-c over e d bi n . Definition 14. Consider a sol ution of nfd for an instan c e ( I , J ) . Fix a bin i ∗ , with u ( i ∗ ) > 0 , and let i ′ b e the smal lest numb er with i ′ > i ∗ such that u ( i ′ ) = 0 , if it exists. We c al l the bin i ∗ wel l-c over e d, if i ′ exists and u ( i ) ≤ 2 d i for al l i = 1 , . . . , i ′ . By O bserv ation 11 and Lemma 13 it can b e shown that w e may assu me that there is at least one we ll-co vered bin in the in stance. Observ ation 15. Consider a solution of nfd for an instanc e ( I , J ) , which c ontains at le ast one fil le d bin. The numb er k of wel l-c over e d bi ns is wel l- define d and we c an assume k ≥ 1 . Pro of. W e ma y assume there is an empt y bin and let i ′ b e the smallest in d ex with u ( i ′ ) = 0. By Observ ation 11 we ma y assume the bin 1 is fi lled and thus i ′ > 1. By Lemma 13 w e ma y assume that for the bins i = 1 , . . . , i ′ w e ha ve u ( i ) ≤ 2 d i . Let k ′ the largest index, suc h that u ( k ′ + 1 ) = 0 and u ( k ′ ) > 0 and u ( i ) ≤ 2 d i for all i = 1 , . . . , k ′ . Let k = { i ∈ { 1 , . . . , k ′ }  u ( i ) > 0 } . Hence k ′ > 1 exists and is w ell-defined as the solution of nfd is unique for a giv en ins tance. It follo ws k is well -defined and k ≥ 1 as u ( 1 ) > 0. W e already can conclude that nfd is at most a 3-appro ximation. Observ ation 16. L et ( I , J ) b e given. If nfd gives a solution with k wel l-c over e d bins, then opt ( I , J ) nfd ( I , J ) ≤ 2 + 1  k . Pro of. Let k ′ b e the largest index of a we ll-co vered bin and let I ′ = { i ∈ { 1 , . . . , k ′ }  u ( i ) > 0 } , b e the set of th e wel l-co v ered bin s . On the one hand we h av e n f d ( I , J ) ≥ ∑ i ∈ I ′ d i and on the other nfd ( I , J ) ≥ kd k ′ . Recall, w e ha ve for ev ery i ∈ I ′ that u ( i ) ≤ 2 d i . Let l b e th e index of th e firs t item, wh ic h nfd could not assign to a bin with index k ′ or smaller. Since u ( k ′ + 1 ) = 0 by definition of k ′ w e fu rther hav e ∑ n j = l s j < d k ′ , otherw ise n fd would hav e filled bin k ′ + 1. It follo w s for th e sum of item sizes s = ∑ n j = 1 s j < ∑ i ∈ I ′ 2 d i + d k ′ . Because opt ≤ s we can b ound op t < ∑ i ∈ I ′ 2 d i + d k ′ ≤ 2 nfd + 1  k ⋅ nfd = ( 2 + 1  k ) nfd . F or a num b er of k ≥ 4 well-co ve red bins we already ha ve the d esired r esu lt. W e no w tur n our atten tion to the cases, wh en k ≤ 3. Lemma 17. L et ( I , J ) b e an instanc e such that n fd gives a solution, in which every fil le d bin is wel l-c over e d and c ontains at most two items. Then opt ( I , J ) nfd ( I , J ) ≤ 2 . Pro of. By Observ ation 11 w e assum e u nfd ( 1 ) > 0. Call a maximal set of neigh b ouring, empt y bins a gap in nf d ’s solution, that is formally a set of b ins { i, i + 1 , . . . , i + i ′ } with u ( i ) = ⋅ ⋅ ⋅ = u ( i + i ′ ) = 0 and u ( i − 1 ) > 0 and i + i ′ + 1 = m + 1 or u ( i + i ′ + 1 ) > 0. Enumerate the gaps from left to right . F or eve ry gap G l consider now the set of filled bins F l b efore this gap, that is, if i ′ is the b in w ith smallest index in G l and i is th e bin w ith highest index in G l − 1 , then F l = { i + 1 , . . . , i ′ − 1 } , w here we assum e 0 is the b in with highest in dex in the “gap” G 0 . 16 If there are t many items on the b ins in F l , th en we mo dify the sets G l in su c h a wa y that  F l  +  G l  = t for ev ery l , if not already the case. If  G l  < t −  F l  , then in tro d uce t −  F l  −  G l  man y b ins with demand d i in to G l , wh ere i is the bin with highest index in G l . If  F l  +  G l  > t then remo ve the  F l  +  G l  − t smallest b ins f rom G l and observe that this is alwa ys p ossible, i. e. we ha ve  G l  ≥  F l  +  G l  − t , since w e hav e  F l  ≤ t . W e r emark, it do es not matter, if  G l  = 0 n o w, for s ome gap l . Observe , b y nfd ’s b eha vior, if I ′ is the set of bins in the mo difi ed instance, w e not only ha ve nf d ( I ′ , J ) = nfd ( I , J ) but a lso all it ems are a ssigned to the same bin s in b oth s olutions, whic h is b eca u se w e copied and remov ed only un filled b ins. Note, if i is an empty bin in nfd ’s solution, and S ≤ i is the set of items, wh ic h reside on a bin w ith ind ex at most i in nfd ’s solution, then nfd can not fill a bin w ith index at most i with only the items from J ∖ S ≤ i . Then, also op t can only fill a bin with ind ex i ′ ≤ i , if there is also an item fr om the set S ≤ i on this bin. Since, if this w ou ld n ot b e the case, then also nfd wo uld hav e filled i , and this argumentatio n h olds for b oth instances ( I , J ) and ( I ′ , J ) . Consider the follo wing relaxed Bin Co ve ring pr oblem. In order to yield the profit d i for a bin i either the bin i has to b e cov ered (with some items) or an item j from th e s et S ≤ i has to b e assigned to i . The problem is clearly a relaxatio n of the ordinary Bin Covering problem. Let opt ∗ denote an optimal algorithm for the relaxed problem. The v alue of this algorithm is obviously only an o ve restimation for op t on the same instance, that is opt ∗ ( I , J ) ≥ opt ( I , J ) . With this mo difi ed notion of o p t we w ill show opt ∗ ( I ′ , J ) ≥ op t ∗ ( I , J ) by construction and then, b ecause of op t ∗ ( I , J ) ≥ opt ( I , J ) , it su ffices to b oun d opt ∗ ( I ′ , J ) ≤ 2 nfd ( I ′ , J ) = 2 nfd ( I , J ) , where we h a ve already exp lained the last equalit y ab o v e. F or opt ∗ ( I ′ , J ) ≥ opt ∗ ( I , J ) w e only h a ve to j ustify that the remo ving of bin s fr om the sets G l do es not m ake opt ∗ lose any pr ofi t on the instance ( I ′ , J ) in comparison to the instance ( I , J ) . Fix a set F l ∪ G l and consider a bin i with an index sm aller than that of an y bin from F l ∪ G l . Let l ′ < l b e suc h that i ∈ F l ′ ∪ G l ′ (b efore mo difying the in stance). On the one han d , recall that no subset of the items in F l ′ + 1 ∪ F l ′ + 2 ∪ . . . can co v er bin i without items in F 1 ∪ ⋅ ⋅ ⋅ ∪ F l ′ . On the other hand , if an item j ′ from a bin fr om F 1 ∪ ⋅ ⋅ ⋅ ∪ F l ′ resides on bin i , then , b y ou r relaxation, opt ∗ already gains the pr ofit for b in i , without assigning another item to this bin . In conclusion opt ∗ w ould not assign suc h an item j ∈ F l ′ + 1 ∪ F l ′ + 2 ∪ . . . to a bin from F l ′ ∪ G l ′ . Note also that assigning suc h an item j to bins w ith larger index than that fr om F l ∪ G l can only b e w orse, by th e ordering of bins by demand. T o s u m up , the pr ofit of all bins in the mo dified instance ( I ′ , J ) w ill b e gained b y opt ∗ with only one item and this is clearly optimal. Hence considerin g a solution of opt ∗ to the instance ( I , J ) and a b in i ∈ I , whic h is n o longer av ailable in the instance ( I ′ , J ) , i. e. i ∉ I ′ , we can mov e all items S i , i. e. the items from bin i in the solution to ( I , J ) , and assign th em to a bin from the set F l ∪ G l in the mo dified instance ( I ′ , J ) . Th us r emo ving bin i do es not h a ve any effect on opt ∗ ’s s olution, since this bin wo u ld b e empt y anyw a y . As this argumen t holds for eve r y bin remo v ed, th e claim opt ∗ ( I ′ , J ) ≥ opt ∗ ( I , J ) follo ws. No w, in order to establish opt ∗ ( I ′ , J ) ≤ 2 nf d ( I ′ , J ) w e claim we can asso ciate ev ery b in from every gap G l to a bin with at least equal demand from F l . This can b e seen as follo ws. 17 If t is the num b er of items on the b ins from F l in nfd ’s solution, then w e hav e  F l  +  G l  = t in the mo difi ed instance and with  F l  ≥ t  2 as at most t wo items reside by p rerequisite on ev ery bin from F l , it follo ws  F l  ≥  G l  . By constru ction every bin in G l has at most as muc h demand as the sm allest bin from F l and the claim is pro v ed. As w e ha ve argued that opt ∗ gains the profi t f or the bins in F l ∪ G l in the mo difi ed in - stance, while nf d gains the profit for the bin s in F l , we h a ve opt ∗ ( I ′ , J ) ≤ 2 nfd ( I ′ , J ) and the statemen t follo ws. Lemma 18. L et ( I , J ) b e an instanc e. If nfd give s a solution with k = 1 wel l-c over e d bins and al l other bins ar e empty, then op t ( I , J ) nfd ( I , J ) ≤ 9  4 . Pro of. W e ma y assume that the well-c o ve red bin cont ains at least three items, otherwise the claim follo ws immediately from Lemma 17. Let t b e the n u m b er of items on bin 1. Observe that if there are t items on bin 1 in nf d ’s solution, we hav e that these ha ve in sum a size of at most t ( t − 1 ) d 1 , w hic h is by the ordering of items and the fact that n fd do es not assign items to already co v ered b in s. With the argumen t that the items t + 1 , . . . , n could not fill an y bins in 2 , . . . , m in nfd ’s solution, it f ollo ws w e hav e ∑ n l = t + 1 s l < d 2 ≤ d 1 . Altoget her we can b ound opt < t t − 1 d 1 + d 1 = ∶ f ( t ) . As f is a monotone decreasing fun ction, it is easy to see f ( t ) ≥ 9  4 d 1 only for 2 ≤ t ≤ 4. As the fu nction give s only an upp er b ound on opt ’s v alue and w e ha ve assumed t ≥ 3 it s u ffices to sh o w that actually opt ≤ 9  4 d 1 for the cases t = 3 and t = 4 in order to establish the statement . If opt wan ts to gain more profit th an t wice the profit nfd gains, then it has to use at least three bins, wh ic h is b ecause nfd fills the largest b in in the ins tance. It is clear our b ound on opt gets only b etter, if th er e are less than three bins in the instance, i. e. m < 3. W e can assume opt uses exactly three bins, if w e do not make an y assu m ptions on their size, b esides that they h a ve at most th e same demand as bin 1 has. F or ease of n otation we relab el the b in s op t us es as 2, 3 and 4 in non-increasing ord er of demand. It suffi ces to sho w that d 2 + d 3 + d 4 ≤ 9  4 d 1 . Clearly , if d 2 + d 3 + d 4 = c and c is fixed, w e can assum e d 2 = d 3 = d 4 = c  3, if w e allo w that opt ma y sp lit the items 4 , . . . , n arbitrarily . This can only b e b etter for opt than an y choic e of d 2 , d 3 , d 4 , since ∑ n l = t + 1 s l < d 4 . Note, that the last b ound holds, sin ce w e h a ve assumed all bins b esides b in 1 are empt y in nfd ’s solution. No w, for the sak e of con tradiction assume w e ha ve d 2 + d 3 + d 4 > 9  4 d 1 . B ecause of d 2 = d 3 = d 4 = c  3 w e hav e that it must b e d 2 , d 3 , d 4 > 3  4 d 1 . F or the case t = 3 we ha v e s 1 + s 2 + s 3 ≤ 3  2 d 1 and w e see that the items 1, 2, 3 do not ev en suffice to fill the b in s d 2 and d 3 . As ∑ n l = t + 1 s l < d 4 w e find that opt cannot fill all three bins and the claim is established. An analogous computation give s a cont radiction for the case t = 4, to o. In ord er to s im p lify the follo wing statemen ts we in tr o duce the term head of the in s tance, whic h is a distinguish ed bin. F or this, fix a solution of nf d to a giv en in stance ( I , J ) . Let i 0 b e the ind ex of the firs t not w ell-co vered bin with u ( i 0 ) > 0 and let i 1 b e the smallest index suc h that u ( i 1 + 1 ) = 0 with i 1 ≥ i 0 . Let i ∗ = max i ∶ u ( i ) > 2 d i { i ≤ i 1 } . Then, the bin i ∗ is called the he ad (of the in s tance). 18 Observ ation 19. Fix a solution of n fd to the instanc e ( I , J ) . If ther e is a fil le d not wel l- c over e d bin, then i ∗ , the he ad of the instanc e, is wel l-define d. Pro of. Recall that i ∗ = max i ∶ u ( i ) > 2 d i { i ≤ i 1 } . W e verify that the set, ov er whic h the m axi- m u m is take n, is non-empt y . By Observ ation 15 w e can assume there exists at least k ≥ 1 w ell-co vered bins in ev ery instance. Ob serv e b y definition that if there is an index i with u ( i ) > 0, such that i is not a well-c ov ered b in, all b in s i ′ with u ( i ′ ) > 0 and i ′ > i are also not well-co ve red. Let i 0 no w b e the s mallest index of a bin with u ( i 0 ) > 0, wh ich is not w ell- co v ered, which exists by precondition. As u ( m ) = 0 as guarantee d b y Ob serv ation 12 there exists a sm allest ind ex i 1 , with i 1 > i 0 and u ( i 1 + 1 ) = 0. By the d efi nition of well- co v erage there has to b e an index i with i 0 ≤ i < i 1 suc h that u ( i ) > 2 d i . No w, let i ∗ ≤ i 1 b e the largest of su c h ind ices. Thus the indices i 0 , i 1 and i ∗ are well-defined and so is the term “head”, if the solution of nf d con tains a not w ell-co vered, non-empt y bin. Lemma 20. L et ( I , J ) b e an instanc e, for which nfd gives a solution with k = 1 wel l-c over e d bin and ther e is a non-empty not wel l-c over e d bin. L et i ∗ b e the he ad of the instanc e. If ther e ar e at le ast thr e e items on bin 1 in nfd ’s solution and o pt assigns at le ast one of these items to a bin with index at le ast i ∗ , then o pt ( I , J ) nfd ( I , J ) ≤ 9  4 . Pro of. Since there is only one w ell-co ve red b in in the instance, it is u ( 2 ) = 0. Let j ′ b e the item on b in i ∗ and recall that s j ′ > 2 d i ∗ , since i ∗ is the h ead of th e instance. Sin ce for ev ery item j from bin 1 w e h a ve s j ≥ s j ′ , we hav e for ev ery suc h item j that it will not only fill a bin with ind ex i ′ ≥ i ∗ , b ut ev en d i ′ ≤ d i ∗ ≤ s j ′  2 ≤ s j  2. Since t ≥ 3 we h a ve that the item with largest index, wh ic h nfd assigns to bin 1, h as size at most d 1  2. By the ordering of items b y size w e ob viously ha ve, if op t assigns at least one of the items from bin 1 to a bin with index i ′ ≥ i ∗ , we can assume it assigns – b esides p ossibly other items – also the item t to s uc h a b in. This can only b etter than c ho osing an item with index smaller than t by the fact that eac h suc h an item will fi ll its resp ectiv e bin i ′ . F or the remaining t − 1 items on bin 1 w e can b ound ∑ t − 1 j = 1 s j < d 1 , b ecause bin 1 was not y et full, when item t was assigned to it. As in Lemm a 18 we can b ound ∑ n j = t + 1 s j < d 1 . Hence assigning at least one of the items to a bin with index i ′ ≥ i ∗ w e can b ound the profit opt yields w ith the ab o ve d iscussion by opt < t − 1  j = 1 s j + s t  2 + n  j = t + 1 s j ≤ d 1 + ( d 1  2 ) 2 + d 1 = 9  4 d 1 , and as nfd ≥ d 1 , th e claim follo ws. No w we giv e a straigh tforward upp er b ound f or nfd in particular for the case there are k = 2 or k = 3 well- cov ered bins and at least one of them con tains at least thr ee items. Lemma 21. L et ( I , J ) b e an instanc e such that nfd give s a solution with k ≥ 2 wel l-c over e d bins. If at le ast one of these bins c ontains at le ast thr e e items then opt ( I , J ) nfd ( I , J ) ≤ 9  4 . 19 Pro of. Note that the case k ≥ 4 is already co vered b y Observ ation 16 and hence w e h a ve only to prov e the cases k = 2 and k = 3. Firstly consid er the case k = 3. Let w. l. o. g. 1, 2 and 3 b e the b ins, to which nfd assigned items. W e ma y assume that if there are t items altogether on the bins 1 , 2 , 3 that there are at least t empt y bins of size at most d 4 in the instance, otherwise w e give enough copies of bin 4 to opt . Recall, if on an y bin i = 1 , 2 , 3 reside three items in nfd ’s solution w e hav e u ( i ) ≤ 3  2 d i and it is clear this term is smaller, if there are more than three items. Because of the ordering and the fact, that all b in s 1, 2 and 3 are w ell-co vered w e can b oun d ∑ t l = 1 s l ≤ 2 d 1 + 2 d 2 + 3  2 d 3 . F ur ther ∑ n l = t + 1 s l < d 4 ≤ d 3 as b in 4 must b e empty , b ecause there are three we ll-co vered bins in the instance. Hence we can b ound the v alue of opt by the sum of item sizes in th e instance op t < 2 d 1 + 2 d 2 + 3  2 d 3 + d 3 = 2 d 1 + 2 d 2 + 5  2 d 3 . F or nfd we yield nfd ≥ d 1 + d 2 + d 3 and hence opt nfd ≤ 2 + 1  2 d 3 d 1 + d 2 + d 3 ≤ 2 + 1  6 ≤ 9  4 . F or k = 2 we ded uce analogously opt ≤ 2 d 1 + 3  2 d 2 + d 2 = 2 d 1 + 5  2 d 2 and nfd ≥ d 1 + d 2 . W e yield opt  nfd ≤ 9  4 . Note, that in particular it do es not matter, if there are bins with index larger than k + 1 that were co v ered b y nfd . It is easy to see that if there is another bin with 3 items or on one bin there are more than three items, the b ound is ev en b etter. Hence th e claim follo ws. Note that Lemma 21 uses a p u re v olume argument. Hence, regarding the p r econditions, it do es not matter, if th ere are add itionally bins in the instance, whic h w ere filled b y nfd . Lemma 22. L et ( I , J ) b e an instanc e such that nfd gi v es a solution with k ∈ { 1 , 2 , 3 } wel l- c over e d bins, su ch that on e ach at most two items r e side. L et the solution further c ontain at le ast one non-e mpty not wel l-c over e d bin, and let i ∗ b e the he ad of the instanc e. Define I ′ = { 1 , . . . , i ∗ } and I ′′ = I ∖ I ′ . F urther let J ′ b e the set of items in nfd ’s solution, which r eside on a bin fr om I ′ , and let J ′′ = J ∖ J ′ . Cal l the set of items op t assigns to a bin fr om I ′ the set A and let B = J ∖ A . Then ( opt ( I ′ , A ) + opt ( I ′′ , B ∖ J ′′ )) nfd ( I ′ , J ′ ) ≤ 2 . Pro of. A t first we observ e nfd ( I ′ , J ′ ) = nfd ( I ′ , J ′ ∪ A ) . Th is is b ecause A ∖ J ′′ ⊆ J ′ and by definition of J ′′ nfd do es n ot assign an y of th e items from J ′′ to a b in from I ′ . In case, B ∖ J ′′ = ∅ , w hic h means op t decides only to us e th e bins f rom I ′ , we can remo ve the bins in I ′′ from th e instance and can compare th e terms from the statemen t on the instance ( I ′ , J ′ ∪ A ) . This instance h as filled not well-c o ve red bins, b ut ob s erv e that these are now the right -most b ins, i. e. there is an in dex i ′ , s uc h that the bins i ′ , . . . ,  I ′  are not w ell-co vered and u ( i ) > 0 for all i ′ ≤ i ≤  I ′  . Hence Observ ation 12 is applicable and th e resulting instance consists only of w ell-co ve r ed b ins. T hen the claim follo ws from Lemma 17. No w assume B ∖ J ′′ ≠ ∅ . Because the smallest item of J ′ resides alone on i ∗ and B ∖ J ′′ ⊆ J ′ w e ha ve ev ery item fr om B ∖ J ′′ will reside alone on a bin from I ′′ . By the same reason, w h en w e add a set I ∗ of b ins with  I ∗  =  B ∖ J ′′  , in whic h eac h bin h as demand d i ∗ , ev ery n ewly in tro duced b in from I ∗ will b e co ve red by an item from B ∖ J ′′ . F urther the bin s in I ∗ are as least as large as the bins in I ′′ . Hence we can mov e eac h item fr om B ∖ J ′′ to the bins in I ∗ and remo ve th e bins from I ′′ , s ince this can only b e b etter for opt . 20 W e argue the pro of of Ob serv ation 12 is still correct in th is situation, ev en if opt has additional bins in the instance. Then we can assume that the bin w ith largest index fr om I ′ is empty in nfd ’s solution. The three thin gs w e ha v e to kee p an eye on are the follo wing. Firstly we h a ve to d elete the same bin s from the instances of n fd and op t . F ur ther we can delete only the bins, whic h are co vered by nfd . A t last, the bins to b e deleted h a ve to b e th e righ t-most ones in the instance, i. e. we ma y only d elete the bins i ′ , . . . , m , if u ( i ) > 0 for all i ≥ i ′ . Then we ma y apply Obs erv ation 12 and hav e that the bin with largest index is empt y in nfd ’s solution. In this situation we can sho w already with Lemma 17 that opt ( I ′ ∪ I ∗ , A ) nfd ( I ′ , A ) ≤ 2. Firstly note it is justified to assume that no other item than the item w e put on i thr ou gh the mo dification of the instance ab o v e is assigned to a bin i from I ∗ . Th is is b ecause opt has chosen this alternativ e to assign su c h an item to a bin f r om I ′′ and we actually ga v e opt at the most add itional profi t. If the add itional bin s in the in stance of opt can take no other items than the items, which already reside on them, then the relaxation carried out in Lemma 17 is still p ossible, w hic h w e ju s tify no w . In th is lemma w e show ed that ev en in a relaxed setting o pt could ac hieve no more than t w ice the profit nfd gained. Ob s erv e that in the r elaxed setting it is only b ett er f or o p t to assign the items no w resid in g on the b in s from I ∗ to the r esp ectiv e bins fr om the F l ∪ G l sets. But th en by Lemma 17 the claim f ollo ws. No w we hav e all to ols at hand to decomp ose a giv en instance, su c h that our pro of ma y restrict to analyze th e sp ecific parts of the decomp osed instance. Lemma 23 (Decomp osition L emma) . L et ( I , J ) b e an instanc e, such that nfd gives a so- lution with k wel l-c over e d bins and at le ast one not wel l-c over e d bin. L et i ∗ b e the he ad of the instanc e . L et J ′ b e the set of items r esiding on the bins 1 , . . . , i ∗ in nfd ’s solution and J ′′ = J ∖ J ′ . F urther let I ′ = { 1 , . . . , i ∗ } and I ′′ = I ∖ I ′ . Then opt ( I , J ) nfd ( I , J ) ≤ max { 9  4 , opt ( I ′′ , J ′′ ) nfd ( I ′′ , J ′′ )} . Pro of. At first we ma y assume that for the num b er of w ell-co vered bins , k , we h a ve k ≤ 3, since otherwise th e claim follo ws b y Ob serv ation 16. Consider the case that all of the w ell-co ve red bins conta in at most t w o items. Observe that nfd ( I , J ) = nfd ( I ′ , J ′ ) + nfd ( I ′′ , J ′′ ) holds for nfd by the d efinition of the s ets. C onsider the solution of o pt . Let A b e the set of items, wh ic h reside on the b ins in I ′ and B th e set of items whic h r esid e on th e bins in I ′′ . Clearly , opt ( I , J ) = opt ( I ′ , A ) + opt ( I ′′ , B ) . W e show that opt ( I ′′ , B ) ≤ opt ( I ′′ , B ∖ J ′′ ) + opt ( I ′′ , J ′′ ) . W e wan t to emphasize, th is is not a trivial relation, since it could b e that opt could not u se all items of the subsets of B , when w e split this s et up. But ob s erv e, her e we ha ve B ∖ J ′′ ⊆ J ′ . Thus all items in B ∖ J ′′ are as least as b ig as the smallest item in J ′ , which was the item on i ∗ . Hence for ev ery item j ∈ B ∖ J ′′ ⊆ J ′ w e hav e s j ≥ 2 d i for eve ry b in i ∈ I ′′ . Thus ev ery suc h item r esides alone on a bin in opt ’s solution for the instance ( I ′′ , B ) (and can also fill su c h a bin) and the inequalit y holds. 21 With this opt ( I , J ) nfd ( I , J ) = opt ( I ′ , A ) + opt ( I ′′ , B ) nfd ( I ′ , J ′ ) + nfd ( I ′′ , J ′′ ) (7) ≤ opt ( I ′ , A ) + opt ( I ′′ , B ∖ J ′′ ) + opt ( I ′′ , J ′′ ) nfd ( I ′ , J ′ ) + nfd ( I ′′ , J ′′ ) (8) ≤ max  opt ( I ′ , A ) + opt ( I ′′ , B ∖ J ′′ ) nfd ( I ′ , J ′ ) , opt ( I ′′ , J ′′ ) nfd ( I ′′ , J ′′ )  (9) ≤ max  9 4 , opt ( I ′′ , J ′′ ) nfd ( I ′′ , J ′′ )  , (10) where we used elemen tary calculus in (9) and in (10) we hav e u sed Lemma 22. No w consider the case that one of the well-c o v ered b ins con tains at least three items. F or k ≥ 2 the claim f ollo ws b y Lemma 21 and we are left to s h o w the claim for k = 1. W e may restrict ourselv es to instances, for w hic h op t assigns all t items, wh ic h reside on bin 1 in nfd ’s sol ution to a b in with index i ′ ≤ i ∗ , as otherwise the claim follo w s b y Lemma 20. Again we decomp ose the s olutions of nfd and opt iden tically as ab o ve, where w e are left to sho w step (10). By the assumption all items from bin 1 are assigned to a b in with index i ′ ≤ i ∗ and as B ∖ J ′′ ⊆ J ′ w e ha ve B ∖ J ′′ = ∅ and hence opt ( I ′′ , B ∖ J ′′ ) = 0. Thus it is enough to show opt ( I ′ , A ) nfd ( I ′ , J ′ ) ≤ 9  4. Again with th e observ ation nfd ( I ′ , J ′ ) = nfd ( I ′ , J ′ ∪ A ) and opt ( I ′ , A ) ≤ opt ( I ′ , J ′ ∪ A ) w e can remo ve with Obser v ation 12 the fi lled not w ell-co vered bins and are left with an in- stance, whic h con tains only one w ell-co vered bin and all other bin s are empt y . T he claim follo ws w ith Lemma 18. Pro of (of Theorem 8). First observe that Example 7 yields a lo wer b ound of 9  4 on th e appro x im ation r atio of nfd . Let k b e the num b er of wel l-co v ered bins in the instance. If k ≥ 4 th en Ob serv ation 16 already giv es the claim. Thus let k ∈ { 1 , 2 , 3 } . If all fi lled bins are also well -co v ered and one of these con tains at least three items, then the claim follo ws from Lemma 18 and Lemma 21. F or the case that ev ery one of the w ell-co vered bins con tains at most t wo items the statemen t follo ws from Lemma 17. No w let there b e a fi lled but not we ll-co v ered bin in the ins tance. Define I ′ = { 1 , . . . , i ∗ } , I ′′ = I ∖ I ′ , J ′ to b e the set of items, whic h are assigned to the bins in I ′ b y nfd and J ′′ = J ∖ J ′ , where i ∗ is the head of the ins tance. No w w e can apply Lemma 23. Observe ( I ′′ , J ′′ ) is a smaller instance, with at least one not we ll-co vered bin less. Hence w e can apply the analysis in a recursiv e step again on this instance. Th e recursion terminates if ( I ′′ , J ′′ ) is an in stance wh ich has only we ll-co vered bins or in which the solution of nfd h as no co ve r ed b ins. Clearly , in the latter case we hav e that nfd is optimal and in the former w e can argue as ab o ve. The algorithm can b e implemente d such that the run ning-time is dominated b y the sorting of the bins and items. 22 Monotonicit y of Next Fit Decreasing for Va riable-Sized Bin Covering F or this subs ection w e introd uce for sak e of shortness the f ollo wing n otion. W e will compare the solutions of nfd to some in stances ( I , J ) and ( I , J ′ ) . F or shortness w e sa y in the instance ( I , J ) a certain prop ert y holds, wh ere w e mean that the solution of nfd to the instance ( I , J ) has this prop ert y . Prop ert y 24. n fd is a monotone algorithm for V ariable-Sized Bin Covering , i. e. if ( I , J ′ ) is an instanc e and J ⊇ J ′ , then it fol low s nfd ( I , J ) ≥ nfd ( I , J ′ ) . Pro of. Obviously it suffi ces to sho w the claim when the in s tance ( I , J ) con tains exactly one new item in comparison to the instance ( I , J ′ ) , i. e. ther e is some j ∈ J with j ∉ J ′ and J = J ′ ∪ { j } . If all b ins i , whic h are filled in the instance ( I , J ′ ) , are filled as w ell in the instance ( I , J ) then the claim follo ws. Th us assume there is a bin i ′ , which is fi lled in the instance ( I , J ′ ) , but is not in th e in stance ( I , J ) . Since J ⊇ J ′ , it then has to b e case th at there is a bin i , which is co v ered in the in stance ( I , J ) , but is not in the instance ( I , J ′ ) . Moreo ve r for the bin with smallest index of these, call it i ∗ , we hav e that i ∗ < i ′ . Th is is immed iately b y the b eh a vior of nf d , since otherwise nfd would h a ve co v ered the bin i ∗ in the ins tance ( I , J ′ ) , to o. Hence we hav e th at for ev ery bin i = 1 , . . . , i ∗ − 1 that either i is co v ered in b oth instances ( I , J ) and ( I , J ′ ) or is not co vered in b oth ins tances. Let j ∗ b e the item with smallest index, whic h r esid es on a b in with index at least i ∗ in the instance ( I , J ′ ) (or wa s not assigned). Since nfd did not co ve r the bin i ∗ in the instance ( I , J ′ ) , we hav e d i ∗ > ∑ n j = j ∗ s j . T h is already gives the claim, as the bins 1 , . . . , i ∗ − 1 w ere iden tically co vered in b oth ins tances and in the instance ( I , J ) additionally at least bin i ∗ is co v ered, wh ic h yields pr ofit d i ∗ and the p rofit, whic h can b e gained on the bins i ∗ , . . . , m in the instance ( I , J ′ ) is smaller as this, as sh o wn. 3.2 Inapp roximabi lit y in the Unit Supply Mo del By r eduction from P ar tition it is not hard to see that the classical Bin Covering is NP - hard and is not approxi mable w ithin a factor of t wo, unless P = NP . This clearly extends to all of the mo dels we consider here. No w the qu estion arises if improv ements in an asymptotic notion, where the optimal pr ofit dive rges, are p ossible. Note that we still require p i = d i , whic h yields that d iv ergence of the optimal profit imp lies div ergence of the total demand of the instance. Ho wev er, it is not obvio u s how to define a suitable asymptotics in the unit supply mo del: If only the total item size div erges, th e optimal profit d o es not. If, in addition, the bin demands (but not their num b er) diverge s , th ese in s tances still con tain P ar tition . Thus w e consider an asym p totics, wh ere the total item size, the total d emand, and the num b er of bins div erges. The follo win g theorem states th at an y algorithm can not ha ve an approximati on ratio of 2 − ǫ , if ǫ > 0 is a constan t, ev en in this case. Eve n stronger, as the choi ce of s = ω ( m ) is p ossible we hav e ρ → 2, for m → ∞ . Theorem 25. Consider V ariable-Sized Bin Covering with unit supply. L et 2 ≤ m ≤ n . Then ther e is an instanc e ( I , J ) with  J  = n + m − 2 , for which an optimal algorithm c overs 23 m bins, but ther e is no p olynomial time algorithm with appr oximation factor b etter than ρ = 2 − m − 2 s / 2 + m − 2 , unless P = NP . Pro of. W e use a red u ction from the P ar tition problem. Recall that f or this we are giv en a set of items P = { 1 , . . . , n } , where item j h as in tegral size s j . Our goal is to find an index set L ⊂ P , such that s ( L ) = s ( P ∖ L ) , i. e. the items fr om P are partitioned in tw o sets of equal size. Let P ′ b e a P ar tition instance and w e refer to the s izes of the items as s ′ j . W e d efine an instance ( I , J ) for V ariable-Sized Bin Covering . W e set I = { 1 , . . . , m } and J = { 1 , . . . , m + n − 2 } w ith s j = 2 s ′ j m for j = 1 , . . . , n and s j = 1 for j = n + 1 , . . . , n + m − 2, i. e. the items of the P ar tition instance are s caled by a factor of 2 m . As m ≤ n this is clearly done in p olynomial time. Recall, s ∶ = ∑ s j and set d 1 = s  2, d 2 = s  2, where we assume s  2 is integral otherwise w e ou tp ut “no”, wh ic h is due to the integral s ′ j (and th u s in tegral s j ). F ur ther we set d 3 = ⋅ ⋅ ⋅ = d m = 1 and d i = p i for all i . No w we s ee that the solution of V ariable-S ized Bin Covering h as a v alue of s + m − 2, if the P ar tition problem has a solution. If the P ar tition problem has no solution, we argue that the v alue of the solution to V ariable-Sized Bin Covering is at most s  2 + m − 2. C onsider fir stly the case that all items 1 , . . . , n are assigned to b ins 1 and 2 b y an algorithm and the items n + 1 , . . . , n + m − 2 are assigned to bins 3 , . . . , m . I n th e non-scaled instance P ′ , for every index set L we ha ve the prop ert y that s ( L ) ≠ s ( P ∖ L ) , i. e. the left-hand su m and the right-hand su m differ by at least one, whic h is b ecause the s ′ j w ere integ ral. Hence in th e instance for V ariable -S ized Bin Covering , w hic h uses the sizes from th e scaled instance P we hav e u ( 1 ) an d u ( 2 ) d iffer for every assignmen t of the items 1 , . . . , n to b ins 1 and 2 by at least 2 m . Let w. l. o. g. b e u ( 1 ) > u ( 2 ) , then we ha v e u ( 1 ) − u ( 2 ) ≥ 2 m , and thus u ( 2 ) ≤ s  2 − m . Consequently , ev en, if all items n + 1 , . . . , n + m − 2 are put by an algorithm on bin 2, we hav e u ( 2 ) ≤ s  2 − m + ( m − 2 ) = s  2 − 2 < s  2 and we see, bin 2 is not co vered, if the items n + 1 , . . . , n + m − 2 are assigned arbitrarily . If the items 1 , . . . n are assigned to arbitrary bins, the v alue of the solution ma y only decrease and we ha ve sho w n th at the v alue of a solution on the giv en instance is at most s  2 + m − 2, if the P ar tition problem has n o s olution. Hence in case an algorithm has appr o ximation r atio smaller than ρ = 2 − m − 2 s / 2 + m − 2 , it can distinguish the cases and solv e the P a r tition problem. 3.3 An A(F) PT AS fo r the Infinite Supply Mo del F or th e classical m o del Csir ik, John son, and Keny on [5] w ere th e fir st to giv e an APT AS. It turn s out that th e ideas of [5] can b e extended for the V ariable-Sized Bin Covering mo del w ith infinite su pply . The basic idea is that s m all b in t yp es can b e ignored without 24 losing too muc h p rofit. Then adjusting the parameters in the algorithm of [5] and adapting the calculatio n s giv es the desired result. After that we can also adapt the m etho d of J an s en and S olis-Oba [11] to impro ve the ru nning time and to obtain an AFPT AS. Here also the LP formulati on has to b e extended appropriately and subroutines ha ve to b e called with appropriately scaled p arameters. W e p ro ve the follo wing theorem in the n ext sections. Theorem 26. Ther e is an AFPT AS for V ariable-Sized Bin Covering in the infinite supply mo del. 3.3.1 An APT AS in the Infinite S upply Mo del It turns out that n ormalizing the demands of bin s is adv antag eous h ere. Thus w e assume in this section 1 = d 1 > ⋅ ⋅ ⋅ > d m > 0. Since we are in the V ariable-Sized Bin Covering mo del w e ha ve d i = p i for all i = 1 , . . . , m . The result of this section will b e the f ollo wing. Theorem 27. Ther e is an APT AS f or the V ariable-Sized Bin Covering pr oblem in the infinite supply mo del. Outline of t he APT AS. Let ǫ > 0 b e the desired appr o ximation factor and w e assume w. l. o. g. th at 1  ǫ is in tegral. I n the algorithm w e delete all b in types with size at most ǫ . This idea was also us ed by Murgolo [13]. Then we sub d ivide th e items into th ree sets: L the set of large items, M the set of medium-sized items, and T the set of tin y items (for a form al definition of L , M , and T , see Algorithm 3.3.1). The large items are fu rther su b divided in to 1  ǫ 4 groups, where eac h group has (almost) equal size with resp ect to th e n u m b er of items it con tains (cf. Step 3 of Algorithm 3.3.1 ). This groupin g tec h nique originates from a pap er of F ernan d ez de la V ega and Lueker [8]. In eac h group, all items are r ou n ded down to the s ize of the s mallest item of the resp ecti v e group. Note, this implies that there are at most k = 1  ǫ 4 man y differen t sizes for the large items. Th e id ea is here that the items of group i can replace the items from grou p i + 1 in an optimal s olution. By this pro cedure only a p rofit b ound ed b y the size of th e fir st grou p is lost. Th en all p ossible assignments – referr ed to as configurations in th e follo wing – of th e large and round ed d o wn items to bins are enumerated. Via an app ropriate LP form u lation a solution is d etermined. More p r ecisely , an LP giv es ho w many bins of eac h type are to b e op ened and according to whic h configuration a bin is assigned items. Here it is crucial that a configuration ma y not fill a bin, bu t the LP form u lation ensu res that only su c h sets of configurations in the solution are u sed so th at the non-large items (i. e. the items from M ∪ T ) can fill the p ossible only p artially co v ered bins in a greedy wa y . A listing of th e algorithm can b e foun d in Figure 3. Definitions for the APT AS. W e introdu ce a set of d efinitions, wh ic h w ill b e h elpful in order to write d o wn the algorithm rather br iefly . • Call a ve ctor v ∈ { 1 , . . . , n } k a configuration, where k = 1  ǫ 4 is an in tegral constan t. L et ℓ = ( ℓ 1 , . . . , ℓ k ) b e the v ector of large sizes, i. e. the size of the items in th e resp ectiv e group. The v alue ℓ i is determined in Step 3 of the Algorithm 3.3.1. • Let e j ∈ { 0 , 1 } k the v ector with an en tr y 1 at p osition j and 0 at all p ositions j ′ ≠ j . 25 • F or a confi guration v i let ℓ ( i ) ∶ = v ⊺ i ⋅ ℓ , i. e. b e the sum of sizes of all items, whic h are con tained in v i . Let n ( i, j ) = v ⊺ i ⋅ e j b e the num b er of items of s ize ℓ j in configuration v i . • Let C j = { i ∈ { 1 , . . . , r }  d j − 1 > ℓ ( i ) ≥ d j } , where d 0 = ∞ , i. e. asso ciat e ev ery configura- tion to a b in t yp e of largest size, suc h that the configur ation co v ers the b in and let C j b e the set of in dices such th at C j con tains all the configur ations asso ciated to bin type j . • Let r ( i, j ) = d j − ℓ ( i ) b e the demand of bin of t yp e j , whic h is not co vered by the configuration i (the remainder). • Let ˜ C j = { i ∈ { 1 , . . . , r }  r ( i, j ) > 0 } , i. e. in ˜ C j there all configu r ations, which do not co v er b in t yp e j . • Let n ( i ) b e the num b er of items of size ℓ i , f or 1 ≤ i ≤ k , in the ins tance. F or the an alysis let opt ( I ; L, T ) d enote the v alue of an optimal algorithm, whic h assigns the items of L and T according to the bin types in I , where it ma y s plit the items in T arbitrarily . No w, we give the key observ ation, whic h lets us adapt the algorithm from [5] to the V ariable-Sized bin covering mo del with in finite supp ly of bins. Observ ation 28. Fix an instanc e ( I , J ) . L et I ′ ∶ = { i ∈ I  d i > ǫ } b e the set of bins, which have demand mor e than ǫ . Then ( 1 + ǫ ) opt ( I ′ , J ) + 1 ≥ opt ( I , J ) . Pro of. C on s ider an optimal solution, in which only bins from the set I ∖ I ′ are co vered, since the b ound is ev en b etter otherwise. W e p artition the items residing on b ins in I ∖ I ′ in t wo sets J 1 and J 2 . In the set J 1 ev ery item h as size smaller than ǫ , in set J 2 ev ery item h as size at least ǫ . O bserve that in an optimal solution the items from J 1 and J 2 will reside on distinct bin s . Hence it suffices to sh o w that for every part i = 1 , 2 of th e instance w e ha ve ( 1 + ǫ ) opt ( I ′ , J i ) ≥ opt ( I , J i ) and w e ma y lose one additional bin in b oth ins tances together. Clearly , if we put all items from J 1 with nfd on bins of size 1, then for every filled bin i w e ha ve u ( i ) < 1 + ǫ . Hence if s ( J 1 ) is th e ov erall size of all items in J 1 w e yield p rofit at least  s ( J 1 )( 1 + ǫ ) ≥ s ( J 1 )( 1 + ǫ ) − 1, wh ilst s ( J 1 ) is an up p er b ound for the p rofit o pt yields on this part of the instance. The items from J 2 w e also assign with nfd to bins of size 1. If the last b in is not filled and so wa s the last b in with the items from J 1 , th en we assign the items from this bin to the bin with the items from J 1 . Hence there is at m ost one not filled bin and it suffices to sho w, we yield a profit of at least a 1 ( 1 + ǫ ) fraction of the profit op t yields on the instance ( I , J 2 ) . Consider a b in i , which is co vered by t many items from J 2 and let S i b e this set of items. If it has fi ll lev el u ( i ) ≤ 1 + ǫ the claim f ollo ws, th us assu me u ( i ) > 1 + ǫ an d let ǫ ′ = u ( i ) − 1. Let I i b e th e set of bin s to wh ic h opt assigned the items from S i and recall that  I i  =  S i  = t , i. e. ev ery item resides alone on its bin in the solution of opt . Since w e ha ve ǫ ′ > ǫ and the last item w as a smallest on bin i , we ha ve for all j ∈ I i that u opt ( j ) − d j ≥ ǫ ′ − ǫ , that is also opt wa sted at least a volume of ǫ ′ − ǫ p er item fr om I i when it had assigned the items from 26 Step 1. Remo ve all bin types with size smaller than ǫ and let m no w b e the num b er of remaining bin t yp es in the instance. Step 2. If n <  s  ǫ 3  +  s  ǫ  , then set L = { 1 , . . . , n } as set of large items and M = T = ∅ as s ets of medium and tin y items. Else order items non-increasingly and define the sets L , M , T of large, medium and tin y items as L = { 1 , . . . ,  s  ǫ 3 } , M = { L  + 1 , . . . ,  L  +  s  ǫ } and T = { L  +  M  + 1 , . . . , n } . Step 3. Sub divide the items of L in k = 1  ǫ 4 groups. Let p =  L  div k and q =  L  mo d k , then the groups 1 , . . . , q ha ve p + 1 items eac h and th e groups q + 1 , . . . , k hav e p items eac h. I n ev ery group i , we round down the size of ev ery item of that grou p to the size s j = ∶ ℓ i , w h ere s j is th e size of the smallest item in the group. Step 4. Enumerate all configurations, s u c h th at we hav e v 1 , . . . , v r are all the configurations and with this compute C j and ˜ C j for j = 1 , . . . , m . Step 5. Introdu ce v ariables y i and z i,j suc h that for 1 ≤ i ≤ r the v ariable y i is asso ciated to configuration i and the v ariable z i,j , with 1 ≤ i ≤ r, 1 ≤ j ≤ m , is asso cia ted to configuration i and bin typ e j . Step 6. Compute s ( T ) and n ( i ) for i = 1 . . . , k . Solv e th e follo wing LP maximize m  j = 1 d j (  i ∈ C j y i +  i ∈  C j z i,j ) (11) sub ject to r  i = 1 n ( i, j )  y i + m  l = 1 z i,l  ≤ n ( j ) j ∈ { 1 , . . . , k } m  j = 1  i ∈  C j r ( i, j ) z i,j ≤ s ( T ) y i ≥ 0 i ∈ { 1 , . . . , r } z i,j ≥ 0 i ∈ { 1 , . . . , r } , j ∈ { 1 , . . . , m } Step 7. Set for every v ariable of the LP y ′ j ∶ =  y j  and z ′ i,j =  z i,j  . Step 8. Construct a solution in the follo wing wa y . a) F or every configuration j = 1 , . . . , r tak e y ′ j man y bins of the asso ciated unique t yp e and fill every bin accordingly to the configuration v i . b) F or every pair ( i, j ) tak e z ′ i,j man y bins w ith demand d j and assign items ac- cordingly to configuration v i . c) Fill the bins created accordingly to the z ′ i,j v ariables in a greedy wa y – for example with nf d – using the items  s  ǫ 2  + 1 , . . . , n , wh ere we are left to show, that this is p ossible. Figure 3: The APT AS. 27 S i to their r esp ectiv e b ins in I i . Since  I i  = t it follo w s that the p r ofit op t gains for ev ery suc h a bin i , which nfd fi lls with u ( i ) = 1 + ǫ ′ > 1 + ǫ , is b ounded by 1 + ǫ ′ − t ( ǫ ′ − ǫ ) = 1 − ǫ ′ ( t − 1 ) + ǫt ≤ 1 − ǫ ( t − 1 ) + ǫt = 1 + ǫ. The profit nf d yields is at least 1, hence, th e claim follo ws. Observ ation 29. We have s ≤ 2 opt ( I , L ∪ T ∪ M ) + 2 . Pro of. W e can assume w. l. o. g. that the largest items in the instance h a ve s ize less th an 1, since otherwise a p repro cessing can remov e larger items and assign them to the bin t yp e with size 1, which is clearly optimal. Then it is easy to see that o pt ( I , L ∪ T ∪ M ) ≥  s  2  , since already nfd giv es such a b ound usin g only the largest bin type w ith demand 1. Rearranging and taking into accoun t the r ounding giv es the claim. Observ ation 30. L et s ≥ 2 and ǫ ≤ 1  6 . Then o pt ( I , L ∪ M ∪ T ) ≤ opt ( I , L ∪ T )( 1 − 2 ǫ ) + 2 . Pro of. T ak e an optimal co vering of the bins with all items, i. e. with items from L ∪ T ∪ M . Since opt yields at most  s  ǫ  man y b ins th e a v erage num b er of large items p er bin is at least 1  ǫ 2 b y the defi nition of the s et L . Hence remo vin g  sǫ  + 1 bins with the largest items, remo ves at least  s  ǫ  many large items. These can now b e used instead of the medium-sized items in the rest of the ins tance, since there are at most so man y medium items in the instance. The mo d ified solution h as at most  s  ǫ  + 1 bins less than the optimal solution. With op t ( I , L ∪ T ∪ M ) ≥  s  2  ≥ s  2 − 1 as argued in th e pro of of Observ ation 29 and as ǫ ≤ 1  6 the r emo v al of the  sǫ  + 1 bins is p ossible, since an optimal solution con tains at least this man y bins. F ur ther w e h a ve sho wn that opt ( I , L ∪ M ∪ T ) −  sǫ  − 1 ≤ opt ( I , L ∪ T ) . With this, opt ( I , L ∪ T ) ≥ op t ( I , L ∪ M ∪ T ) −  sǫ  − 1 ≥ opt ( I , L ∪ M ∪ T ) − sǫ − 1 ≥ opt ( I , L ∪ M ∪ T ) − 2 ǫ opt ( I , L ∪ T ∪ M ) − 2 ǫ − 1 ≥ ( 1 − 2 ǫ ) opt ( I , L ∪ M ∪ T ) − 2 ǫ − 1 , where we hav e u s ed Observ ation 29 in the third line. Rearranging and usin g ǫ ≤ 1  6, the claim follo ws. Observ ation 31. If ǫ ≤ 1  6 then opt ( I , L ∪ M ∪ T ) ≤ 1 + ǫ 1 − 2 ǫ opt ( I ′ ; L, T ) + 4 . Pro of. W e hav e opt ( I , L ∪ M ∪ T ) ≤ ( 1 + ǫ ) opt ( I ′ , L ∪ M ∪ T ) + 1 (12) ≤ ( 1 + ǫ )  1 1 − 2 ǫ opt ( I ′ , L ∪ T ) + 2  + 1 (13) ≤ 1 + ǫ 1 − 2 ǫ opt ( I ′ ; L, T ) + 4 , (14) 28 where (12) is by Ob serv ation 28, (13) is b y Observ ation 30, and (14) is by the precondition ǫ ≤ 1  6 and the observ ation opt ( I ′ , L ∪ T ) ≤ opt ( I ′ ; L, T ) . Observ ation 32. L et ǫ ≤ 1  10 . Consider the solution ac c or ding to o pt ( I ′ ; L, T ) . If L ′ denotes the se ts of lar ge items, which is obtaine d by r ounding down the item sizes fr om L as done in Step 3 of Algo rithm 3.3.1 , then opt ( I ′ ; L, T ) ≤ 1 − 2 ǫ 1 − 4 ǫ − 2 ǫ 2 opt ( I ′ ; L ′ , T ) + 9 . Pro of. By the roun ding p ro cedure in Step 3 w e ha v e that in a solution, in wh ic h the items from L w ere round ed down, an item from a group i can replace an item from the group i + 1, where we lose the p + 1 largest items or p largest items, if  L  can b e d ivid ed by k . Hence, if L ′ denotes the set of rounded large items, w e ha v e opt ( I ′ ; L, T ) ≤ opt ( I ′ ; L ′ , T ) + p + 1. With this, since p ≤  s  ǫ 3  ⋅ ǫ 4 b y the n u m b er of groups, we hav e opt ( I ′ ; L, T ) ≤ opt ( I ′ ; L ′ , T ) + sǫ + 2. Hence, w e can b ound opt ( I ′ ; L, T ) ≤ opt ( I ′ ; L ′ , T ) + s ǫ + 2 (15) ≤ opt ( I ′ ; L ′ , T ) + 2 ǫ opt ( I , L ∪ M ∪ T ) + 2 ǫ + 2 (16) ≤ opt ( I ′ ; L ′ , T ) + 1 + ǫ 1 2 ǫ − 1 opt ( I ′ ; L, T ) + 10 ǫ + 2 (17) ≤ opt ( I ′ ; L ′ , T ) + 1 + ǫ 1 2 ǫ − 1 opt ( I ′ ; L, T ) + 3 , (18) where we used Observ ation 29 in (16), Obs erv ation 31 in (17) and the fact th at ǫ ≤ 1  10 in step (18). Rearran ging terms and using again ǫ ≤ 1  10 giv es the claim. Pro of (of Theorem 27). Assu me ǫ ≤ 1  10. W e fi rstly show that all bin s the algorithm outputs are fi lled, then w e b ound the appro ximation ratio. As argued in Ob serv ation 32, we can obtain a solution for the original problem f r om the solution with the large items, wh ich w er e round ed do wn in Step 3 of Algorithm 3.3.1. Hence it suffices to sh ow that in Step 7 all op ened bins are filled. The bins in whic h items fr om configurations associated to v ariables y ′ j are filled b y d efinition of the y j v ariables. F or bin s corresp onding to the z ′ i,j v ariables we argue that th ese are filled in Step 7 (c). By the size of the smallest b ins an y solution has at most  s  ǫ  m an y bins. Sin ce nfd do es n ot assign items to filled bins , we ha ve that at most  s  ǫ  items assigned by nfd do not co ver any demand and we sa y the su m of item sizes is w asted. Since nfd w astes at the worst the largest  s  ǫ  of the items in M ∪ T , we hav e that at most a sum of sizes of s ( M ) is wasted, by definition of the set M . Since s ( T ) is at most the demand to co ver as en forced by the constraints of the LP , and s ( M ∪ T ) is the su m of the items sizes, whic h is a v ailable to n f d in order to fill the not co vered bins induced by the z ′ i,j v ariables, we ha ve nfd can fill all those bins. W e no w giv e the calculation for th e approximat ion factor, wher e we explain the steps thereafter. Let I ′ = { i ∈ I  d i > ǫ } . W e ha ve 29 opt ( I , L ∪ M ∪ T ) ≤ 1 + ǫ 1 − 2 ǫ opt ( I ′ ; L, T ) + 4 (19) ≤ 1 + ǫ 1 − 2 ǫ  1 − 2 ǫ 1 − 4 ǫ − 2 ǫ 2 opt ( I ′ ; L ′ , T ) + 9  + 4 (20) ≤ 1 + ǫ 1 − 4 ǫ − 2 ǫ 2 opt ( I ′ ; L ′ , T ) + 17 (21) ≤ 1 + ǫ 1 − 4 ǫ − 2 ǫ 2   m  i = 1 d i    c ∈ C i y c +  c ∈  C i z i,c     + 17 (22) ≤ 1 + ǫ 1 − 4 ǫ − 2 ǫ 2   m  i = 1 d i    c ∈ C i y ′ c +  c ∈  C i z ′ i,c   + 1  ǫ 4   + 19 (23) (24) In (19) we use O bserv ation 31. In (20) we app ly Observ ation 32. (21 ) uses th e fact that ǫ ≤ 1  10. (22) is easy to observe and fi nally , in (23) w e r ound down the v ariables of the LP , whic h is explained as follo ws. W e hav e that our LP has only 1 + 1  ǫ 4 constrain ts b esides th e n on-negativit y constraints. Hence an optimal basic solution has at most 1 + 1  ǫ 4 fractional v alues and hence w e lose at most so many bin s with d emand 1 du e to roun ding do wn the fractional v ariables. As suc h a solution can b e found in p olynomial time, since the LP h as p olynomial size in n (though exp onen tially in 1  ǫ , whic h is a constant) the b ound follo ws. If 1 + ǫ ′ > 1 is the desired appro x im ation r atio we set ǫ = ǫ ′  13 and run our algorithm w h ic h give s an appr o ximation ratio of at least 1 + ǫ ′ min u s a constan t term. Also observe, b y our choice of ǫ , and ǫ ′ ≤ 1 w. l. o. g., the assumption ǫ ≤ 1  10 w as justified. 3.3.2 An AFPT AS in the infinite supply mo del Jansen and S olis-Oba [11] gav e an AFPT AS for the Bin Covering problem. In this section w e extend their metho d to work for V ariable-sized Bin Covering in the infinite su pply mo del in order to pro ve Th eorem 26. F ormulation a s a resource sharing p roblem and overall m etho d. The AFPT AS do es not solv e the linear program (11) (LP ) in Step 6 of the APT AS exactly . Instead w e approximate LP (11). W e will sho w later, ho w to tran s form this solution into a feasible solution for LP (11). Then we ap p ly the roundin g p ro cedure form Theorem 27. Recall that k = 1  ε 4 is the n um b er of d ifferen t large s izes and r = n k = n 1 / ε 4 is the n um b er of confi gurations. Let x = ( y 1 , . . . , y r , z 1 , 1 , . . . , z 1 ,m , z 2 , 1 , . . . , z 2 ,m , . . . , z r, 1 , . . . , z r,m ) b e a solution ve ctor to the V ariable-Sized Bin Covering p roblem. W e restate LP (11 ) in the follo wing form. 30 λ ∗ = min  λ  (25) r  i = 1 n ( i, j ) n ( j )  y i + m  l = 1 z i,l  ≤ λ 1 ≤ j ≤ k, x ∈ B t m  j = 1  i ∈  C j r ( i, j ) s ( T ) z i,j ≤ λ x ∈ B t  , where B t =  x  m  j = 1 d j     i ∈ C j y i + m  l = 1  i ∈  C j z i,l    = tε and ∀ i, l ∶ y i ≥ 0 , z i,l ≥ 0  Note that for λ = 1 th e constraint s of LP (11) are equiv alent to th e constrain ts of L P (25). The v alue t defining the simplex B t thereb y will b e set su ch that tε is the (appr oximate ) v alue of an optimal solution and we can guess t via binary searc h . W e explain this in more detail later. Sup p ose tε is the true v alue of an optimal int egral solution. Then λ ∗ = 1 is the optimal v alue of the resource sharing p roblem and the corresp onding solution vect or x giv es also a solution to LP (11). Jansen and Solis-Oba give in [11] a solution to LP (25) f or the case when m = 1 an d d 1 = 1 with th e pric e dir e ctive de c omp osition metho d [9, 12] and sho w how this can b e transformed int o a ( 1 + ε ) -approximat ion for the V ariable-Sized Bin Co vering problem. W e can extend th eir tec hniqu e to work for m bin t yp es. LP (25) is a c onvex blo ck- angular r esour c e sharing pr oblem . Resource sharing p r oblems can b e solv ed with the pr ice-directiv e decomp ositio n metho d [9, 12] w ithin an y giv en approx- imation factor. W e give a br ief o ve rview of this metho d. An algorithm for solving a r esource sharing prob lem fin ds a solution iterativ ely . It starts with an arb itrary feasible solution x ∗ and determines a pr ice v ector p = ( p 1 , . . . , p k + 1 ) , wh ose comp onent s are n on-negativ e. It requires to solve a subp roblem, called th e b lo ck pr o gr am , whose solution dep end s on p . W e will state the blo c k program for our p roblem b elo w. A linear combinatio n of an optimal solution ˆ x to the b lo c k program and the p r evious solution x ∗ found b y th e price-directiv e decomp osition metho d so far d etermines an up d ated solution x ∗ for the original resource sharing problem. After a certain n um b er of iterations for any giv en δ > 0 the pr ice-directiv e decomp osition metho d guaran tees a solution x ∗ with ob jectiv e v alue at most ( 1 + δ ) λ ∗ , wh ere λ ∗ is the ob jectiv e v alue of an optimal solution for the resource sharing pr ob lem. W e are left to show ho w to transform this solution into a ( 1 + ε ) -appro ximate solution for the V ariable -sized Bin Covering p roblem. Statement of the blo ck program. The b lo c k program w e ha ve to solv e is min { p T Ax  x ∈ B t } , (26) 31 where matrix A = ( a j,i ) den otes the ( k + 1 ) × r ( 1 + m ) coefficient matrix corresp ond ing to the constraint s of LP (25). W e give the en tries of A in more detail. Let f ( j ) = ( j − r ) d iv m and g ( j ) = ( j − r ) mo d m . Then a j,i =                                n ( i, j ) n ( j ) if 1 ≤ j ≤ k and 1 ≤ i ≤ r n ( i ′ , j ) n ( j ) if 1 ≤ j ≤ k and r + 1 ≤ i ≤ r ( 1 + m ) , w here i ′ = f ( i − 1 ) + 1 0 if j = k + 1 and 1 ≤ i ≤ r r ( i ′ , j ′ ) s ( T ) if j = k + 1 and r + 1 ≤ i ≤ r ( 1 + m ) , where i ′ = f ( i − 1 ) + 1 and j ′ = g ( i − 1 ) + 1 . Observe , the co efficien ts in columns 1 ≤ i ≤ r are the co efficien ts of the y i v ariables and the co efficien ts in columns r + 1 ≤ j ≤ r ( 1 + m ) are the coefficients for the v ariables z i ′ ,j ′ , with i ′ = f ( i − 1 ) + 1 and j ′ = g ( i − 1 ) + 1. Note that we n eglecte d here for ease of presentat ion that some additional en tr ies a j,i for j = k + 1 and r + 1 ≤ i ≤ r ( 1 + m ) ma y b e zero, namely if i ∉ ˜ C j , i. e. if configur ation i co v ers bin t yp e j . Since B t is a simp lex, an optimal solution x ∗ for p rogram (26) will b e attai ned at a v ertex. That is one comp onent of x ∗ has v alue tε and all other comp onents are zero. Hence an optimal solution corresp onds to a sin gle confi guration. Thus it is enough to find a configuration with smallest price, in order to s olv e the blo c k p roblem, wher e the price of a confi guration is determined by p T A as follo ws. Let 1 ≤ i ≤ r b e a configuration. R ecall, that we hav e m + 1 v ariables for configur ation i , whic h are the v ariables y i , z i, 1 , . . . , z i,m . Eac h of these v ariables was asso ciat ed to a bin t yp e j . The v ariable z i,j for 1 ≤ j ≤ m w as asso ciated to bin t yp e j and the v ariable y i w as asso ciated to a bin type with largest demand , wh ic h is co v ered b y i . Fix no w one of th ese m + 1 v ariables and sa y j ′ is the ind ex of this v ariable in the solution ve c- tor x = ( y 1 , . . . , y r , z 1 , 1 , . . . , z 1 ,m , z 2 , 1 , . . . , z 2 ,m , . . . , z r, 1 , . . . , z r,m ) . Note that by fixing a v ariable also a bin t yp e j is fixed and also th e other wa y round. W e defi n e the price of configuration i with resp ect to bin type j as p T Ae j ′ , w here e j ′ again denotes the v ector w ith a one in ro w j ′ and zero otherwise. That is, the pr ice of configuration i with resp ect to bin t yp e j is deter- mined b y multi plying the pr ice v ector p with column j ′ in the matrix A . L et p = ( p 1 , . . . , p k + 1 ) . Then the pr ice of a configuration i with resp ect to bin type j is ∑ k l = 1 n ( i, l ) p l  n ( l ) , if config- uration i co ve r s a bin of t yp e j or ∑ k l = 1 n ( i, l ) p l  n ( l ) + r ( i, j ) p k + 1  s ( T ) if it do es not co v er a bin of t yp e j . As argued it is enough to find a configuration i with smallest pr ice with resp ect to s ome bin t yp e j . Define t wo typ es of inte ger programs (IP) τ i, 1 = min k  l = 1 p l n ( l ) u l (27) s. t. k  l = 1 ℓ l u l ≥ d i u l ∈ { 0 , . . . , n ( l )} 32 and τ i, 2 = min k  l = 1 p l n ( l ) u l + p k + 1 d i − ∑ k l = 1 ℓ l u l s ( T ) (28) s. t. k  l = 1 ℓ l u l ≤ d i . u l ∈ { 0 , . . . , n ( l )} Here the v ariables u l denote the n u m b er of items of size t yp e l to choose. Hence it is not hard to see that IP (27) fi nds a c h eap est confi gu r ation, wh ich co v ers b in t yp e j and IP (28) finds a c heap est configuration, whic h do es not cov er a bin t yp e j . Hence taking the o ve rall cheapest configur ation, i. e. th e configuration wh ic h gives the minim u m v alue in th e set M = { τ j, 1 , τ j, 2  1 ≤ j ≤ m } , is the configur ation whic h is the solution to the blo ck problem. Solution of the blo ck p roblem. In the pr evious section w e redu ced the prob lem of fi nding an optimal s olution to th e blo c k problem to fin ding the configu r ation, whic h giv es th e minim um v alue of the set M and in this s ection w e sho w , h o w to find it by solving I Ps (27) and (28). IP (27) is the minimum kn apsac k problem and it is folklore th at there exists a FPT AS for it. By a dynamic pr ogram and an app ropriate roundin g tec h nique Jansen and Solis-Oba [11] can also obtain an FPT AS for the program min { τ 1 , 1 , τ 1 , 2 } , wh ere d 1 = 1. W e can u s e th eir FPT AS as a pro cedure in order to find the o v erall c heap est configuration, i. e. for m d ifferen t and arbitrary d i v alues. F or this w e scale the constrain ts app ropriately: Let ℓ = ( ℓ 1 , . . . , ℓ k ) b e th e vecto r of all item s izes and p = ( p 1 , . . . , p k + 1 ) the p rice v ector. Let ℓ ( d i ) = ( ℓ 1  d i , . . . , ℓ k  d i ) and p ( d i ) = ( p 1 , . . . , p k , d i p k + 1 ) . W e replace the co efficien ts in the constraint s of IPs (27) and (28) by the corresp onding co efficien ts fr om the ve ctors p ( d i ) and ℓ ( d i ) . W e observ e that u = ( u 1 , . . . , u k ) is a solution to the p rogram τ ′ i, 1 = min k  l = 1 p l n ( l ) u l (29) s. t. k  l = 1 ℓ l d i u l ≥ 1 , u l ∈ { 0 , . . . , n ( l )} if and only if u is a solution to IP (27). Note further that the resp ectiv e ob jectiv e v alues of the solutions are iden tical in b oth problems, since all scaled v alues d o n ot contribute to the ob jectiv e fun ction. Hence a solution u of IP (27) with ob jective v alue τ ′ i, 1 is a solution of u of IP (29) w ith iden tical ob jectiv e v alue. Similarly we conclude that th e program 33 τ ′ i, 2 = min k  l = 1 p l n ( l ) u l + d i p k + 1 1 − ∑ k l = 1 u l ℓ l  d i s ( T ) (30) s. t. k  l = 1 ℓ l d i u l ≤ 1 . u l ∈ { 0 , . . . , n ( l )} has a solution u with ob jectiv e v alue τ ′ i, 2 if and only if u is a solution to IP (28) with the same ob jectiv e v alue. Note that IP (29) and IP (30) are of the s h ap e of IPs (27) and (28), where d i = 1. Hence th e FPT AS of J ansen and S olis-Oba for th e b lo c k p roblem is app licable in this setting. Ove r all we ha ve foun d an algorithm for solving the problem min { τ i, 1 , τ i, 2 } : divide the item sizes in ℓ by d i and multiply the ( k + 1 ) -st comp onen t of the p rice vec tor p with d i and compu te a solution of the blo c k pr ob lem with the mo dified size and price vect or with the FPT AS of Jansen and Solis-Oba in [11]. As argued the confi gu r ation m inimizing min { τ i, 1 , τ i, 2 } o ver all bin t yp es i = 1 , . . . , m is a ( 1 + ε ) -appro ximate solution for the b lo c k problem of V ariable-Sized Bin Co ve ring . App roximation guarantee and running time analysis. Lemma 33. A solution to LP (11) with obje ctive value at le ast ( 1 − 2 ε ) opt − O ( 1  ε 4 ) and length O ( 1  ε 4 ) c an b e found in p olynomial time. Pro of. As in the p ro of Theorem 27 we assu m e w. l. o. g. that d 1 = 1 and that the size of eac h item is smaller than 1. Then ob viously opt ( I , J ) = opt ≤ n . Since we wan t to find an asymptotic FPT AS w e ma y assume that opt ≥ 1. W e partition th e interv al [ 1 , n ] int o subinterv als of size ε . Because opt ≥ 1 w e kn o w there exists a ˆ t suc h that ( 1 − ε ) opt ≤ ˆ tε ≤ opt . F or giv en t let λ ( t ) b e the v alue of an optimal solution to LP (25) and λ ∗ ( t ) b e the v alue of the solution to L P (25) giv en b y the p rice directiv e d ecomp osition metho d. If tε ≤ o pt then λ ( t ) ≤ 1, since λ ′ = 1 is the v alue of a solution, when tε = opt . In th is case the price directiv e decomp osition metho d fin ds a solution with λ ∗ ( t ) ≤ 1 + ε . If the price directiv e decomp osition metho d fi n ds a solution with v alue λ ∗ ( t ) > 1 + ε we know that ther e is no solution with v alue λ ( t ′ ) ≤ 1 for an y t ′ ≥ t . Hence by binary search we fin d a largest t ∗ suc h that λ ( t ∗ ) ≤ 1 + ε : As argued there exists a ˆ t su ch th at ( 1 − ε ) opt ≤ ˆ tε ≤ opt . Since for ev ery t ′ ≤ ˆ t a solution with v alue λ ( t ′ ) ≤ 1 + ε can b e f ound by the price d irectiv e decomp osition m etho d we find a t ∗ ≥ ˆ t . Thus ( 1 − ε ) opt ≤ t ∗ ε . Also, the solution v ector x ∗ corresp ondin g to the solution with v alue λ ∗ ( t ∗ ) ma y not b e feasible, n amely if λ ∗ ( t ∗ ) > 1. W e can transform the solution v ector x ∗ in to a solution x ′ b y m ultiplying eac h coord inate by the v alue 1 − ε . It is easy to see that if x ∗ is a solution for LP (25), suc h that th e left-hand side of eac h constraint has v alue λ ∗ ≤ 1 + ε , then for the solution x ′ in LP (25) the left-hand side of eac h constrain t has v alue at most ( 1 − ε ) λ ∗ ≤ ( 1 − ε )( 1 + ε ) ≤ 1 − ε 2 ≤ 1. Hence x ′ is a feasible solution for LP (11). As argued th e ob jectiv e 34 v alue λ ∗ ( t ∗ ) ≥ ( 1 − ε ) and hen ce the ob jectiv e v alue λ ′ for the scaled solution x ′ is at least ( 1 − ε ) 2 ≥ 1 − 2 ε . A solution x ′ ma y ha ve up to O (( k + 1 )( ε − 2 + ln ( k + 1 ))) co ord inates, since there are so man y calls to the blo ck solv er by the price directiv e decomp osition metho d [12]. W e can transform this solution x ′ in to a basic solution with at most 1 + 1 / ε 4 fractional co ordin ates in ord er to impro ve the app ro ximation ratio. This can b e done by s olving a homogeneous linear system of equ alities, cf. [11] for d etails. W e argue ab out the ru nning time. An algorithm for a resource sharing pr oblem with M constrain ts give n by Jans en and Zh ang [12] fin ds a solution in O ( M ( ε − 2 + ln M )) iterations and has an o v erhead of O ( M ln ln ( M / ε ) ) op erations p er step. In our case it is M = k + 1. The d ynamic pr ogram of J ansen and Solis-Oba in [11], wh ic h w e use as a pro cedu r e for solving the blo ck pr oblem has a ru nning time of O ( n 2 / ε ) p er call. The o v er h ead for scaling the price ve ctor b efore we call this program is O ( k ) and as w e ha ve m calls to th is program w e need time O ( m ( k + n 2 / ε )) in order to solve the blo c k pr oblem. Note that, in particular, neither the r unnin g time of the blo ck solv er nor of the algorithm from [12] d ep end s on th e size of ∣ B t ∣ = O ( n 1 / ε 4 ) . W e need an add itional time of O (( k + 1 )( ε − 2 + ln ( k + 1 )) M ( 2 + k )) for transforming the solution ve ctor with O (( k + 1 ) ( ε − 2 + ln ( k + 1 ))) co ord inates in an vecto r w ith O ( 1 + k ) co- ordinates, w here M ( 2 + k ) is the runn ing time for solving a homogeneous linear system of 2 + 1 / ε 4 equations in 2 + k v ariables. Pro of (of Theorem 26) . The AFPT AS works iden tically as the APT AS, with the exce p tion that w e appro ximate LP (11) as giv en by Lemma 33. W e first argue ab ou t the appr o ximation guaran tee. W e can pr o ceed as in the pr o of of Th eorem 27. After In equalit y (22 ) we hav e to tak e into accoun t that LP (11) is only app ro ximated. Th en we can b ound the add itional loss of bins b y the roundin g p ro cedure of the APT AS as done in In equalit y (23). 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