Optimization with Demand Oracles

Optimization with Demand Oracles Ash winkumar Badanidiyuru Departmen t of C omputer Science Cornell Un v ersit y ashwin85@cs .cornell.edu Shahar Dobzinski Departmen t of Computer Science Cornell Unv e rsity shahar@cs.c ornell.edu Sigal O ren Departmen t of Computer Science Cornell Unv e rsity sigal@cs.co rnell.edu No v em b er 7, 2018 Abstract W e study c ombinatoria l pr o cur ement auctions , where a buy er with a v aluation function v a nd budget B wishes to buy a set of items. Each item i has a co st c i and the buyer is interested in a set S that maximizes v ( S ) sub ject to Σ i ∈ S c i ≤ B . Spec ia l cases of combinatorial pro cur e ment auctions are classic a l pr oblems from submo dular o ptimization. In particular, when the cos ts are all e q ual ( c ar dinality c onstr aint ), a classic result by Nemhauser et a l shows tha t the g reedy algorithm provides an e e − 1 approximation. Motiv ated by many pap er s that utilize demand queries to elicit the preferences o f agents in economic s e ttings, we develop alg orithms that guara n tee improved approximation ra tios in the presence of demand ora cles. W e are able to break the e e − 1 barrier : we pres ent a lgorithms that use only p olynomia lly many demand quer ies and hav e approximation ratios of 9 8 + ǫ for the general pro blem a nd 9 8 for maximiza tion sub ject to a cardinality constra int . W e a lso cons ider the more general class of suba dditive v aluations . W e present alg orithms that obtain an approximation ratio of 2 + ǫ for the genera l pr oblem and 2 for max imization sub j ect to a cardinality co nstraint. W e gua rantee these a pproximation ratios even when the v aluations are non- monotone. W e show that these ratios ar e essentially o ptimal, in the s e ns e that for a ny consta nt ǫ > 0, obtaining an approximation ratio of 2 − ǫ r equires exp onentially many demand queries. 1 In tro duc tion W e study the follo wing c ombinatorial pr o cur ement auction problem: a buye r with a v aluation v and budget B w ishes to pu rc hase a set of items S , where eac h item i has a cost c i . The buy er is in terested in maximizing his v alue v ( S ) while n ot o ve rs p endin g (Σ i ∈ S c i ≤ B ). T ruthful mec hanisms for v arious v ariations of this pr ob lem were stud ied in [26] and its f ollo wup [9]. In this pap er we study the p roblem fr om a p ure combinatorial optimization p oin t of view. T his is analogous to the com bin atorial auctions literature, where one branc h stud ies incen tiv es issues (e.g., [8, 20, 6 , 2]), and the other stud ies the problem as a p ure combinatoria l optimization problem ignoring incentiv es (e.g., [22, 16 , 27, 7, 11, 13, 10]). W e fo cu s on cases where the v aluation fun ction v is su bmo du lar (for eac h t wo sets S and T , v ( S ) + v ( T ) ≥ v ( S ∪ T ) + v ( S ∩ T )) or subadd itiv e (for eac h tw o sets S and T , v ( S ) + v ( T ) ≥ v ( S ∪ T )). Submo dular v aluatio ns capture cases where the buye r exhibits decreasing marginal u tilities, an d subadditive v aluations capture complement freeness. See, e.g., [22], for a more elab orate discus s ion. When v is submo du lar a combinato rial pr o curement auction is a reform ulation of a classical problem fr om submo dular optimization, maximization sub j ect to a knapsack constrain t: eac h item i has a cost c i and the goal is to find a maxim um-v alue set S suc h that Σ i ∈ S c i ≤ B , for a giv en budget B . A sp ecial case is maximization su b ject to a cardinalit y constrain t: find a s et S of s ize k with the highest v alue. Man y other generalizations and v arian ts of this pr oblem we re studied (e.g., [14, 27, 17, 21, 18, 5]). W e wan t ou r algorithms to run in time p olynomial in n , the n umb er of items. Ho w ev er, the v aluati on function is an ob ject of size 2 m . The com binatorial auctions literature therefore us ually assumes that access to the v aluatio n is done via an oracle. Tw o t yp es of queries were extensively studied: value queries (giv en a set S , return v ( S )) and demand queries (giv en pr ices p 1 , . . . , p m return a set S su c h that S ∈ arg max T v ( T ) − Σ j ∈ T p j ). Here we use demand queries to solv e the combinato rial pro curement auctions problem. This p ath w as already tak en in [9 ], b ut let us men tion t wo reasons: • Economic interpretation: man y algorithms f or economic settings, either truthfu l or not truthful, assume that the v aluatio ns are accessed via demand oracles (e.g., [13, 11, 7, 8, 1, 20]). As w as argued extensivel y in the literature (e.g., [4 , 3, 25, 19, 24]) demand queries are a natural w a y f or agen ts to express their preferences. • Bypassing imp ossibilit y results: it is known that p olynomially man y value queries cannot guaran tee an approximati on ratio of n 1 2 − ǫ ev en for optimization s ub ject to a cardinalit y constrain t if the v aluation of the buyer is subadd itiv e [26, 7] 1 . Hence we must us e stronger oracles to ac hiev e reasonable approximat ion ratios. The classical result in subm o dular optimizatio n sho ws that the greedy algorithm pro vides an appro ximation ratio of e e − 1 [14] for optimizati on su b ject to a cardinalit y constraint, and th at this is the b est p ossible with a p olynomial num b er of v alue queries [23, 12]. W e show that d emand queries allo w us to b reak the e e − 1 barrier: Theorem: There exists a 9 8 -appro ximation algorithm f or the pr oblem of maximizing a monotone submo d ular function sub ject to a cardinalit y constraint that make s a p olynomial num b er of demand queries. F or the problem of maximizing a monotone submo d ular function su b ject to a knapsac k 1 In fact this b oun d holds even if the v aluation is fractionally subadd itive – a v aluation is fractionally subadditive (a.k.a. XOS) if it is the maximum of several additive v aluations. A formal defin ition will b e presented later. 1 constrain t, for every constant ǫ > 0, there exists a ( 9 8 + ǫ )-appro ximation algorithm that makes a p olynomial num b er of demand queries. W e start by p resen ting a natural linear p rogram. Th e du al of the linear p rogram has exp onen tial man y constrain ts bu t only p olynomially many v ariables, so we sho w that th e dual can b e solv ed via the ellipsoid metho d, using a d emand query as a separation oracle 2 . W e show v arious structural prop erties of optimal solutions to the LP , e.g., there are at most tw o sets in their s u pp ort. W e n o w face an additional c hallenge: in general, none of these sets (or a p art of them, in case they violate the card in alit y constrain t) pro vides a go o d enough appr o ximation. The key step is showing that by taking one set and augmenti ng it us in g the other set we get a new com bined set th at do es pr o vide the sp ecified appro ximation ratio. Next, we consider the class of sub additiv e v aluations. In [9] a (2 + ǫ ) app ro ximation was obtained for maximization sub ject to a cardinalit y constraint. On top of sligh tly imp ro ving this ratio, w e presen t th e first constant- appr o ximation algorithm f or maximization sub ject to a knapsac k constrain t, improving o v er the O (log m ) ratio in [9]: Theorem: There exists a 2-approximat ion algorithm for the problem of maximizing a subad d itiv e function su b ject to a cardinalit y constrain t that mak es a p olynomial num b er of demand queries. F or the prob lem of maximizing a subadditive fun ction s u b ject to a knapsack constraint, for ev ery constan t ǫ > 0, there exists a (2 + ǫ )-appro ximation algorithm that mak es a p olynomial n umber of demand queries. These algorithms do not assu me that the v aluation is monotone. In particular, th e algorithms guaran tee the sp ecified appro ximation ratio also for n on-monotone submo d ular v aluations (a class that receiv ed muc h atten tion recen tly – see, e.g., [21, 18, 15, 5]). In fact, we once again break the lo w er b ound for approximat ion with v alue queries: it is kno wn that a 2 . 03-appro ximation algorithm for maximizing non-monotone submo du lar function m us t use exp onentia lly m an y v alue q u eries [15]. W e also show ho w to ob tain purely com b in atorial algorithms via a certain t yp e of a natural “ascending auction”, p ossibly with an additional augmen tation step. An exciting direction is to analyze these auctions and similar ones from a more economic p oin t of view, although we do not push this direction f u rther in th is pap er. Our b ound s hav e another implication: they enable u s to constru ct a “monotone estimator” (in the language of [9]) and impro v e the b est known approximat ion ratio of a deterministic truthful mec hanism for m aximizing su badditive v aluatio n sub j ect to a knapsack constrain t to O (log 2 m ). The b est previously kn o wn b ound for this setting wa s O (log 3 m ) [9]. Are our b ound s optimal? W e p ro vide an example with a matc hing inte gralit y gap of 9 8 for optimizing monotone submo dular v aluations. F or non-monotone sub m o dular v aluatio ns we present an example with a matc hing inte gralit y gap of 2. Moreo v er, we show that f or sub additiv e v aluatio ns our results are optimal (in f act, they are optimal ev en if the fun ction is kn o wn to b e fractionally subadditive) : Theorem: Fi x some constan t ǫ > 0. Let A b e a (p ossibly randomized) (2 − ǫ )-appr o ximation algorithm for maximizing a fr actionally su badditiv e function sub j ect to a cardinalit y constraint. Then, A mak es exp onenti ally many demand qu er ies. W e n ote that proving limits on the p o wer of demand queries requires develo pin g significan t amount of nov el mac hinery 3 . The idea is to start w ith some sp ecific v aluation v , and obtain a v aluation v T 2 This first step is very similar to the use of demand oracles in algorithms for combinatorial auctions, where demand oracles are u sed to solv e the LP [7 , 11 , 13], but t he similarit y b etw een th e algorithms end s here. 3 The most relev ant low er b ound that applies sp ecifically to d emand queries is th at of Nisan and S egal [24]. 2 b y “plant ing” some bundle T of size k with high v alue. W e would like to sh o w that determining whether such plan ting o ccurred requires an exp onen tial num b er of demand queries. Th e c hallenge here is that a single d emand query can v erify wh ether the v aluation is v T for many bu ndles T sim ultaneously . Ne verthele ss, we sho w that the p o wer of a single demand query is limited: for ev ery demand query there exists a set of items (relativ ely small but of significan t size) that is con tained in all bun dles T that the demand query sim ultaneously verifies. This enables us to upp er b ound the n umb er of b undles that can b e sim ultaneously verified by a single d emand query , and suffices to deriv e the theorem. W e also u se this tec hnique to rule out an FPT AS for m aximizing a submo d ular function sub ject to a knapsac k constrain t. Op en Questions In this pap er we initiated the systematic study of optimization with demand queries. Let us no w men tion sev eral intrig u in g questions that w e lea v e op en. The first obvious one is to d etermine the p ossible approximat ion ratio when the v aluation of the b uy er is su bmo du lar, b oth in the monotone and in the n on-monotone cases. It is also in teresting to u nderstand the p ossib le approximat ion ratios u s ing differen t constrain ts. Examples include optimizing s u bmo du lar fun ctions sub ject to matroid constrain t [14, 27], multiple knapsac ks [17, 18] and their combinatio n [21 , 5]. What is th e appro ximation ratio p ossible if demand q u eries are a v aila ble in all th ese settings? W e mak e a fir s t step in this direction b y providing an O ( k ) approxima tion algorithm for optimizing subadditiv e function su b ject to k kn apsac k constrain ts. W e do not know neither whether this is the optimal ratio p ossible nor how to constru ct go o d algorithms for the other settings. P ap er O rganization Section 3 p resen ts the LP for the problem and pr o v es some structural prop erties of optimal solutions. In S ection 4 w e giv e our 9 8 -appro ximation algorithms for su bmo du lar v al u ations. S ection 5 pro vides appro ximation algorithms for subadditiv e v aluat ions. W e sh o w that the algorithms for su b additiv e v aluati ons are in fact optimal in Section 6 . The pro of that there is no FPT AS f or submo du lar v aluati ons sub ject to a knapsac k constraint is also in that section. 2 Preliminaries V aluations and Problems Definition Let M b e a set of items, | M | = m . Let v : 2 M → R b e a set function. W e assume that v ( ∅ ) = 0. v is monotone non-de cr e asing if for eac h S ⊆ T , v ( S ) ≤ v ( T ). In this pap er w e are mainly interested in th e f ollo win g t wo problems 4 : in the maximization subje ct to c ar dinality c onstr aint prob lem w e are giv en a num b er k , and we wan t to find the largest v alue set S , | S | = k . In the maximization subje ct to budget c onstr aint pr oblem we are giv en a b udget B and a cost c i for eac h item i . Th e goal is to find the largest-v alue set S , su c h that Σ j ∈ S c j ≤ B . W e sometimes use the name maximization subje ct to a knapsack c onstr aint for this problem. W e consider th is problem und er v arious restrictions on the v aluati ons. W e sa y that v is submo d- ular if for every S, T ⊆ M we ha v e that v ( S ) + v ( T ) ≥ v ( S ∪ T ) + v ( S ∩ T ). v is sub ad ditive if for ev ery S, T ⊆ M we ha v e that v ( S ) + v ( T ) ≥ v ( S ∪ T ). A v aluati on is additive if v ( S ) = Σ j ∈ S v ( { j } ). Notice that eve ry su bmo du lar v aluation is also su badditiv e, and ev ery additive v aluatio n is submo d ular. How ev er, the settin g of [24] is m uch simpler, and their ideas do not seem to b e applicable in ou r case. 4 W e also consider some generalizations, which will b e d efined in the prop er su b sections. 3 Another typ e of v aluat ions that w e consid er is fractionally s u badditive (or X OS). A v aluation is XOS if there exist additive v aluatio n s v 1 , . . . , v l suc h that v ( S ) = max i v i ( S ) (w e say that i is a maximizing clause of S ). It is kno wn [22] that eve ry X OS v aluatio n is su b additiv e, and that every monotone su bmo du lar v aluation is X OS, and that there are sub ad d itiv e v aluatio ns that are not X OS, and XOS v aluations th at are not subm o dular. W e sometimes use th e notation v ( S | T ) to d en ote v ( S ∪ T ) − v ( T ). F or the problem of maximizing a fu n ction sub j ect to bud get constrain t, w e sometimes u se the notation C ( S ) = Σ j ∈ S c j . Demand Queries Giv en prices p 1 , . . . , p m , a demand query retur ns a b undle S , S ∈ arg m ax T v ( T ) − Σ j ∈ T p j . T h e set of bun dles that ac hiev e the max is called th e demand set of the query . Ou r algorithms pro vide the guaran teed approximati on ratio without making any assumption ab out whic h sp ecific bu ndle is returned from the demand set. F or low er b oun ds, as p oin ted out in [24], some un natural tie- breaking rules ma y su pply u nrealistic information ab out the v aluation. Therefore, as in [24], our lo w er b ou n d assumes any tie-breaking ru le that do es not dep en d on the v aluation, e.g., return the lexicographicall y-fir st b u ndle in the demand s et. Another t yp e of w ell-studied query is value query : giv en a set S , return v ( S ). It is kno wn [4] th at a v alue query can b e simulate d b y p olynomially many demand queries (but exp onen tially man y v alue qu eries ma y b e required to simula te a sin gle demand query). Th is pap er concen trates in designing algorithm w ith p olynomially many d emand queries, so we freely assume that we ha v e access to v alue queries. 3 The Structure of the L P Our algorithms are based on findin g goo d r oundings of fractional solutions. These fractional so- lutions are obtained by solving a natural linear relaxation of our problems. Th e k ey ingredient is analyzing th e structur al prop er ties of optimal fractional solutions: w e give limits on the n umb er of elemen ts in the su pp ort, analyze th eir costs, and giv e p recise formulas for the w eight s of the elemen ts in th e supp ort. The LP can b e solv ed u sing demand oracles. Moreo v er, we show that there is a combinato rial metho d of obtaining solutions to the L P , using a p ro cess that can b e view ed as an ascending auction. The LP is presented for the m ore general case of maximization sub ject to a b udget constrain t (maximizatio n su b ject to a cardinalit y constraint is a sp ecial case wher e for eac h item i , c i = 1 and B = k ). Maximize: P S x S · v ( S ) Subje ct to: • P S x S · C ( S ) ≤ B . • P S x S ≤ 1. • F or eac h bundle S : x S ≥ 0. Although the LP has an exp onen tial num b er of v ariables, w e sh o w that it can still b e solv ed using a p olynomial n umb er of d emand queries: Prop osition 3.1 The LP c an b e solve d using a p olynomial numb er of demand qu e ries. 4 Pro of: Consider th e dual of the pr imal LP: Minimize: y · B + z Subje ct to: • F or eac h bundle S : z + y · C ( S ) ≥ v ( S ). • y ≥ 0. • z ≥ 0. W e use the ellipsoid metho d to solv e the d ual LP and th us obtain a solution to the pr imal LP . F or the ellipsoid metho d to w ork we need to imp lemen t a separation oracle that finds some set S that violates the constraint y · C ( S ) + z ≥ v ( S ). T o do that, consider th e demand q u ery dq = y · ( c 1 , . . . , c m ), and let T b e the b undle that dq returns. If z ≥ v ( T ) − y · C ( T ) then by the definition of demand qu ery f or ev ery other set S w e ha v e z ≥ v ( T ) − y · C ( T ) ≥ v ( S ) − y · C ( S ). A key ingredien t of our algorithms is the follo win g structural prop ert y of the LP . Definition 3.2 A fr actional solution is a strict solution if ther e ar e two variables x S 1 and x S 2 , such that C ( S 1 ) ≤ B , C ( S 2 ) > B , x S 1 = α , and x S 2 = 1 − α , wher e α = C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) . Prop osition 3.3 Either ther e exists an optimal inte gr al solution to the LP or ther e exists an optimal strict solution. Given an optimal solution to the LP, we c an tr ansfor m it in p olyno mial time to a fr actional solution with the same value that is either inte g r al or strict. Pro of: Supp ose that for eac h set S in the su p p ort of the solution we hav e that C ( S ) ≤ B . In this case we claim that there is an in tegral solution, s ince max( v ( S ) | x S > 0) ≥ P S x S v ( S ). Consider n o w the other extreme case, where for eac h set S in the supp ort of the solution w e ha v e that C ( S ) > B . In th is case we sho w that there is a fractional solution with exactly one v ariable in the supp ort: X S x S v ( S ) = X S x S X j ∈ S c j v ( S ) P j ∈ S c j = max( v ( S ) P j ∈ S c j | x S > 0) · X S x S X j ∈ S c j ≤ max( v ( S ) P j ∈ S c j | x S > 0) · B Let S 2 = arg max( v ( S ) P j ∈ S c j | x S > 0). In this case we set S 1 = ∅ . Observe that by setting α = C ( S 2 ) − B C ( S 2 ) w e get a strict fractional solution with v alue at least equal to the v alue of the optimal f ractional solution. The only case that is left to handle is the case where there are tw o sets with x S 1 , x S 2 > 0 and C ( S 1 ) ≤ B and C ( S 2 ) > B . W e sho w a strict fractional solution that is op timal. By complementary slac kness we h a v e that z + y · C ( S 1 ) = v ( S 1 ) and z + y · C ( S 2 ) = v ( S 2 ). Solving for z and y we get z = v ( S 1 ) C ( S 2 ) − v ( S 2 ) C ( S 1 ) C ( S 2 ) − C ( S 1 ) and y = v ( S 2 ) − v ( S 1 ) C ( S 2 ) − C ( S 1 ) . Hence the v alue of the optimal fractional solution is v ( S 1 )( C ( S 2 ) − B )+ v ( S 2 )( B − C ( S 1 )) C ( S 2 ) − C ( S 1 ) . Consider now th e strict fr actional solution x S 1 = C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) , x S 2 = B − C ( S 1 ) C ( S 2 ) − C ( S 1 ) . Its v alue is exactly the v alue of the optimal f r actional solution: C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) · v ( S 1 ) + B − C ( S 1 ) C ( S 2 ) − C ( S 1 ) · v ( S 2 ) 5 W e now present a combinatoria l m etho d of obtaining strict fractional solutions to the LP . An imp ortant adv an tage of th is metho d is that it uses only uniform-pr ice qu eries (unlike the previous LP-based approac h that requires arbitrary demand queries). W e m ak e use of the follo wing simple prop erty of demand qu eries: Lemma 3.4 L et p = ( p 1 , ..., p m ) . L et S 1 b e some bu nd le that maximizes the pr ofit in pric e s λ 1 · p . L et S 2 b e some bund le that maximizes the pr ofit in pric es λ 2 · p . Then λ 1 ≤ λ 2 if and only if Σ j ∈ S 1 p j ≥ Σ j ∈ S 2 p j . Pro of: By the definition of demand quer y w e ha v e v ( S 2 ) − λ 1 Σ j ∈ S 2 p j ≤ v ( S 1 ) − λ 1 Σ j ∈ S 1 p j and v ( S 1 ) − λ 2 Σ j ∈ S 1 p j ≤ v ( S 2 ) − λ 2 Σ j ∈ S 2 p j . Ad ding the t w o in equalities and simplifying w e get ( λ 1 − λ 2 )(Σ j ∈ S 2 p j − Σ j ∈ S 1 p j ) ≥ 0. Th is implies the lemma. Definition 3.5 λ ≥ 0 is the b oundary if ther e exists S 1 , S 2 , C ( S 1 ) ≤ B , C ( S 2 ) > B , such that S 2 is in the demand set in pric es λ · ( c 1 , . . . , c m ) and for every smal l enough δ > 0 we have that S 1 is in the demand set in pric es ( λ + δ ) · ( c 1 , . . . , c m ) . We say that S 1 and S 2 ar e two b ound ary bun d les . Let us f u rther explain the notion of b ound ary bund les. F or simplicit y , th is paragraph considers only the simpler cardinalit y constraint. Consider some pr ices λ · (1 , . . . , 1). When λ = 0, th e most profitable bund le consists of all items. Lemma 3.4 implies that as the v alue of λ in cr eases, the size of bund les in the demand set d ecreases (ev ent ually , wh en λ is b ig enough, the demand set will b e empt y). The b oundary is the sp ecific λ for whic h the demand set in λ · (1 , . . . , 1) con tains only bund les that violate the card inalit y constrain t, bu t for ( λ + ǫ ) · (1 , . . . , 1) the demand set conta ins bund les that resp ect the cardinalit y constraint. The b oundary bundles (and λ ) can b e found by an ascending auction in w hic h we con tinously 5 increase the v alue of λ and obtain a profit-maximizing bun dle in th e curr en t prices. The b oundary is the λ for which the prices λ · ( c 1 , . . . , c m ) are the sup rem um of the p oints where su pply exceeds demand. W e we re surp rised to find that the b oundary bund les d efine a high-v alue fractional solution: Lemma 3.6 L et S 1 and S 2 b e two b oundary bund les and let λ b e the b oundary. L et α = C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) . Then x S 1 = α and x S 2 = 1 − α is a strict solution to the LP. Mor e over, let O b e the optimal inte gr al solution. Then, x S 1 v ( S 1 ) + x S 2 v ( S 2 ) ≥ v ( O ) . Pro of: T o see that the solution resp ects the b udget constrain t: α · C ( S 1 ) + (1 − α ) · C ( S 2 ) = C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) · C ( S 1 ) + B − C ( S 1 ) C ( S 2 ) − C ( S 1 ) · C ( S 2 ) = B ( C ( S 2 ) − C ( S 1 )) C ( S 2 ) − C ( S 1 ) = B As for the second part, S 1 and S 2 are in the demand set of their resp ectiv e prices, therefore: v ( O ) − ( λ + δ ) · C ( O ) ≤ v ( S 1 ) − ( λ + δ ) · C ( S 1 ) v ( O ) − λ · C ( O ) ≤ v ( S 2 ) − λ · C ( S 2 ) 5 Of course, to obt ain a discrete pro cess one may increment λ in some discrete amount. This results in some loss in the approximation ratios of t he algorithms th at we construct (the loss dep en ds on the size of the increments). 6 Multiplying the first inequalit y b y α , the second inequalit y by (1 − α ), and sum ming u p the t wo w e get that (using B ≥ C ( O )): v ( O ) − ( λ + δ ) B + αδ · B ≤ α · v ( S 1 ) + (1 − α ) v ( S 2 ) − α · ( λ + δ ) · C ( S 1 ) − (1 − α ) · λ · C ( S 2 ) = α · v ( S 1 ) + (1 − α ) v ( S 2 ) − C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) · ( λ + δ ) · C ( S 1 ) − B − C ( S 1 ) C ( S 2 ) − C ( S 1 ) · λ · C ( S 2 ) = α · v ( S 1 ) + (1 − α ) v ( S 2 ) − λ · B − C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) · δ · C ( S 1 ) Rearranging b oth sides we get: v ( O ) − δ · B + αδ · B ≤ α · v ( S 1 ) + (1 − α ) v ( S 2 ) − C ( S 2 ) − B C ( S 2 ) − C ( S 1 ) · δ · C ( S 1 ) No w when we let δ go to 0 we get that x S 1 v ( S 1 ) + x S 2 v ( S 2 ) ≥ v ( O ), as needed. 4 A 9 8 -Appro ximation for Monotone Su bmo du lar V aluations As we ha ve shown, an y s trict solution to the LP with card inalit y constraint con tains t wo bun dles, a “large” bund le with more than k items, and a “small” bund le with at most k items. A natural approac h for an appro ximation algorithm is to select the maxim um of the f ollo win g t wo bu ndles: the “small” bu n dle, or an high-v alue c h un k of size k fr om the “large ” bun dle. Unfortu n ately , th ere are examples that show that this approac h cannot pro vide an appro ximation ratio b etter than 2. Hence, our s trategy is subtler: w e start with the small bund le and grab as m uc h v alue as w e can from the large bu ndle. W e sho w that this combined bun dle pro vides an approximati on ratio of 9 8 . W e also extend this algorithm to handle m aximizatio n sub ject to a knapsac k constrain t. Here the approximat ion ratio we get is sligh tly worse: 9 8 + ǫ . W e en umerate o v er all sets of high-cost items, then use a v ariation of our algorithm for a cardinalit y constrain t as an algorithm for in stances with only lo w-cost items. W e complemen t these results by presenting an in s tance w ith an int egralit y gap of 9 8 (in the app end ix), ev en for maximization sub ject to cardinalit y constraint. Theorem 4.1 The fol lowing algorithms exist: • A 9 8 -appr oximation algorithm for the pr oblem of maximizing a monotone su b mo dular function subje ct to a c ar dinality c onstr aint that makes a p olynomia l numb er of demand queries. • A ( 9 8 + O ( ǫ )) -appr oximation algorithm for the pr oblem of maximizing a monotone submo dular valuation sub je ct to a knapsack c onstr aint that makes a p olynomial numb er of demand queries, for any fixe d ǫ > 0 . W e start with the algorithm for the cardin alit y constraint. W e first pr esen t the algorithm, then commen t on h o w it can b e efficien tly implement ed. The Algorithm 1. Obtain a strict solution to the LP: x S 1 = α and x S 2 = 1 − α . Let k 1 = | S 1 | and k 2 = | S 2 | . 2. Find S ′ ⊆ S 2 , | S ′ | = k suc h that v ( S ′ ) ≥ k k 2 v ( S 2 ). 7 3. Find S ′′ ⊆ S 2 − S 1 , | S ′′ | = k − k 1 , su c h that v ( S ′′ | S 1 ) ≥ k − k 1 | S 2 − S 1 | v ( S 2 | S 1 ). 4. Outp u t max( v ( S 1 ∪ S ′′ ) , v ( S ′ )). As for th e efficien t implemen tation of the algorithm, w e hav e already argued in Pr op osition 3.3 that an optimal and strict solution to the LP can b e found with a p olynomial num b er of demand qu eries. Alternativ ely , Step 1 can also b e imp lemen ted com binatorially as an ascending auction: start with a price p er item of 0 and increase it gradu ally . When supply exceeds demand, consider the b ound ary bun dles, and obtain a strict solution as in Lemma 3.6. Steps 2 and 3 can b e implemen ted using th e follo wing folklore lemma (see Lemma 4.5 for a pro of of a more general setting). Lemma 4.2 L et v b e a submo dular valuation, S b e some bund le, and let t ≤ | S | b e some inte ger. Then, ther e exists a set S ′ ⊆ S , | S ′ | = t such that v ( S ′ ) ≥ t | S | v ( S ) . In addition, S c an b e found using a p olynomial numb er of value querie s. Step 2 follo ws immediately from Lemma 4.2. Step 3 follo ws by observing that the marginal v aluati on v ( ·| S 1 ) is s ubmo d ular to o, and applying Lemma 4.2 again. W e are left with proving the appro ximation ratio of the algorithm: Lemma 4.3 L et A b e the bu nd le that the algorithm outputs. Then, v ( A ) ≥ 8 9 ( x S 1 v ( S 1 ) + x S 2 v ( S 2 )) . Pro of: Recall that v ( S ′′ | S 1 ) ≥ k − k 1 | S 2 − S 1 | v ( S 2 | S 1 ). That is (also using | S 2 − S 1 | ≤ k 2 ): v ( S ′′ ∪ S 1 ) − v ( S 1 ) ≥ k − k 1 k 2 ( v ( S 2 ∪ S 1 ) − v ( S 1 )) Rearranging and using v ( S 2 ∪ S 1 ) ≥ v ( S 2 ) (since v is monotone) we ha ve: v ( S ′′ ∪ S 1 ) ≥ k − k 1 k 2 v ( S 2 ) + k 1 + k 2 − k k 2 v ( S 1 ) (1) W e are finally r eady to p r o v e the appro ximation ratio: α · v ( S 1 ) + (1 − α ) · v ( S 2 ) = α k 2 k 1 + k 2 − k  k 1 + k 2 − k k 2 v ( S 1 ) + k − k 1 k 2 v ( S 2 )  +(1 − α − α k − k 1 k 1 + k 2 − k ) v ( S 2 ) ≤ α k 2 k 1 + k 2 − k v ( S 1 ∪ S ′′ ) + (1 − α − α k − k 1 k 1 + k 2 − k ) v ( S 2 ) ≤ α k 2 k 1 + k 2 − k v ( S 1 ∪ S ′′ ) + (1 − α − α k − k 1 k 1 + k 2 − k ) k 2 k v ( S ′ ) ≤ ( α k 2 k 1 + k 2 − k + (1 − α − α k − k 1 k 1 + k 2 − k ) k 2 k ) max( v ( S 1 ∪ S ′′ ) , v ( S ′ )) ≤ γ · max( v ( S 1 ∪ S ′′ ) , v ( S ′ )) where the second inequalit y is due to (1). It remains to b oun d th e v alue of γ (the pr o of can b e found in the app endix): Claim 4.4 L et γ = max k 1 ≤ k ǫ · a } . (c) F or the set of items M L,L ′ , obtain a s tr ict solution to the LP w.r.t. v L,L ′ and budget B L,L ′ = B − C ( T ): x L,L ′ S 1 = α L,L ′ and x L,L ′ S 2 = 1 − α L,L ′ . (d) Find S ′ ⊆ S 2 , C ( S ′ ) ≤ B L,L ′ suc h that v L,L ′ ( S ′ ) ≥ B L,L ′ (1 − ǫ ) C ( S 2 ) v ( S 2 ). I f there is no su ch S ′ let A L,L ′ = ∅ and con tinue to the n ext iteration. (e) Find S ′′ ⊆ S 2 − S 1 , C ( S ′ ) ≤ B L,L ′ − C ( S 1 ), su ch that v L,L ′ ( S ′ ) ≥ ( B L,L ′ − C ( S 1 ))(1 − ǫ ) C ( S 2 − S 1 ) v L,L ′ ( S 2 ). If there is no suc h S ′ let A L,L ′ = ∅ and con tinue to the n ext iteration. (f ) Let A L,L ′ = arg max( v ( T ∪ S 1 ∪ S ′′ ) , v ( T ∪ S ′ )). 2. Outp u t arg max L,L ′ v ( A L,L ′ ). Notice that if ǫ is constan t then there are only p olynomially man y iterations. S in ce eac h iteration needs p olynomially many demand queries, the total num b er of demand queries is p olynomial to o. W e will use Lemma 4.5 to imp lement Steps (1d) and (1e), similarly to the algo rithm for a cardinalit y constrain t. (Notice th at v L,L ′ is a s ubmo d u lar v aluat ion.) Lemma 4.5 L et v b e some sub mo dular valuation, and S b e some bund le. Th en, ther e exi sts a chain S 1 ⊆ . . . ⊆ S | S |− 1 ⊆ S | S | = S , wher e | S t | = t and v ( S t ) ≥ C ( S t ) C ( S ) v ( S ) for eve ry t . Pro of: W e prov e the existence of S t for t = | S | − 1. The lemma will then follo w as it im p lies that w e can find S 1 ⊆ S 2 ⊆ . . . ⊆ S | S |− 1 ⊆ S | S | = S , wh ere | S t | = t , and for eac h l , | S l | + 1 = | S l +1 | and v ( S l ) ≥ C ( S l ) C ( S l +1 ) v ( S l +1 ). No w we will hav e that v ( S t ) ≥ C ( S t ) C ( S t +1 ) v ( S t +1 ) ≥ C ( S t ) C ( S t +1 ) · C ( S t +1 ) C ( S t +2 ) · . . . · C ( S | S |− 1 ) C ( S | S | ) v ( S ) = C ( S t ) C ( S ) v ( S ) Notice th at giv en v ( S l ) w e can v ( S l − 1 ) by considerin g the l su bsets of size l − 1 and taking the one with the highest v alue. W e no w pr o v e the lemma for t = | S | − 1. W e hav e that v ( S ) = | S | X j =1 v ( { j }|{ 1 , . . . , j − 1 } ) ≥ | S | X j =1 v ( { j }| S − { j } ) where the inequalit y holds b ecause v is s ubmo d ular and h as decreasing marginal u tilities. In particular we ha ve that f or some item j , v ( { j | S − { j }} ) ≤ c j C ( S ) , i.e., v ( S − { j } ) ≥ C ( S −{ j } ) C ( S ) v ( S ). Let S ′ = S − { j } . T his completes th e pr o of of the lemma. 9 W e only analyze one particular iteration, wh en L is th e set of the 1 ǫ 2 items with the h ighest costs in th e optimal solution. F urthermore, let a 1 , ..., a | L | b e some ord er on the items of L so th at for ev ery i , v ( a i |{ 1 , ...., a i − 1 } ) ≥ v ( a i +1 |{ 1 , ...., a i } ). Suc h order can b e obtained b y starting fr om the empty set and greedily taking the item from L with the highest marginal con tribution th at has not b een tak en yet . Let L ′ = { a (1 − ǫ ) | L | , . . . a | L | } . Observe that by su bmo du larit y the v alue of the optimal un restricted solution in M L,L ′ ∪ T is at least (1 − ǫ ) O P T , where OPT is the v alue of the optimal solution. Also n otice that for eac h item j ∈ M L,L ′ w e ha v e that c j ≤ ǫB L,L ′ since B L,L ′ > | L ′ | · c j . F or S tep (1d) note that for some S t w e ha ve by Lemma 4.5 that C ( S t ) ≤ B L,L ′ and v L,L ′ ( S ′ ) ≥ B L,L ′ (1 − ǫ ) C ( S 2 ) v ( S 2 ). Since for eac h item j ∈ M L,L ′ w e h a v e that c j ≤ ǫB L,L ′ w e immed iately get C ( S t ) ≥ B L,L ′ (1 − ǫ ) and S tep (1d) follo ws. Step (1e) f ollo ws by observing that the marginal v aluati on v L,L ′ ( ·| S 1 ) is submo d ular too and ap p lying Lemma 4.5 similar to S tep (1d). Lemma 4.6 γ 1 − ǫ v L,L ′ ( A L,L ′ ) ≥ x L,L ′ S 1 v L,L ′ ( S 1 ) + x L,L ′ S 2 v L,L ′ ( S 2 ) . Pro of: Recall that v L,L ′ ( S ′′ | S 1 ) ≥ ( B − C ( S 1 ))(1 − ǫ ) C ( S 2 − S 1 ) v L,L ′ ( S 2 | S 1 ). That is (also u sing C ( S 2 − S 1 ) ≤ C ( S 2 )): v L,L ′ ( S ′′ ∪ S 1 ) − v L,L ′ ( S 1 ) ≥ ( B − C ( S 1 ))(1 − ǫ ) C ( S 2 ) ( v L,L ′ ( S 2 ∪ S 1 ) − v L,L ′ ( S 1 )) Rearranging and using v L,L ′ ( S 2 ∪ S 1 ) ≥ v L,L ′ ( S 2 ) (since v is monotone) we hav e: v L,L ′ ( S ′′ ∪ S 1 ) ≥ ( B − C ( S 1 ) C ( S 2 ) v L,L ′ ( S 2 ) + C ( S 1 ) + C ( S 2 ) − B C ( S 2 ) v L,L ′ ( S 1 ))(1 − ǫ ) (2) α · v L,L ′ ( S 1 ) + (1 − α ) · v L,L ′ ( S 2 ) = α C ( S 2 ) C ( S 1 ) + C ( S 2 ) − B ( C ( S 1 ) + C ( S 2 ) − B C ( S 2 ) v L,L ′ ( S 1 ) + B − C ( S 1 ) C ( S 2 ) v L,L ′ ( S 2 )) + (1 − α − α B − C ( S 1 ) C ( S 1 ) + C ( S 2 ) − B ) v L,L ′ ( S 2 ) ≤ α C ( S 2 ) C ( S 1 ) + C ( S 2 ) − B v L,L ′ ( S 1 ∪ S ′′ ) 1 − ǫ +(1 − α − α B − C ( S 1 ) C ( S 1 ) + C ( S 2 ) − B ) v L,L ′ ( S 2 ) ≤ α C ( S 2 ) C ( S 1 ) + C ( S 2 ) − B v L,L ′ ( S 1 ∪ S ′′ ) 1 − ǫ +(1 − α − α B − C ( S 1 ) C ( S 1 ) + C ( S 2 ) − B ) C ( S 2 ) B v L,L ′ ( S ′ ) 1 − ǫ ≤ ( α C ( S 2 ) C ( S 1 ) + C ( S 2 ) − B + (1 − α − α B − C ( S 1 ) C ( S 1 ) + C ( S 2 ) − B ) C ( S 2 ) B ) · max( v L,L ′ ( S 1 ∪ S ′′ ) , v L,L ′ ( S ′ )) 1 − ǫ ≤ γ 1 − ǫ · m ax( v L,L ′ ( S 1 ∪ S ′′ ) , v L,L ′ ( S ′ )) where γ sho we d to b e at most 9 8 in Claim 4.4. Recall that w e lost at most ǫ · O P T by discarding items in L ′ . W e therefore h a v e that th e v alue of the solution is at least 8(1 − ǫ ) 9 (1 − ǫ ) · O P T . 10 5 A 2 -Appro ximation for Su badditiv e V aluations W e s h o w that there exists a 2-appro ximation algorithm for maximization sub ject to cardin ality constrain t. This is the b est ratio achiev able with a p olynomial num b er of demand qu eries eve n if the v aluation is X OS, as w e sh o w in S ection 6. While this is only a sligh t im p ro ve ment o v er the (2 + ǫ ) approximati on algo rithm f or the setting of [9], we then show ho w to extend this algorithm to pro vide a (1 + k 1 − ǫ )-appro ximation for maximization su b ject to k -knapsac k constrain ts. In particular this implies that there exists a (2 + O ( ǫ ))-approximat ion algorithm for maximization sub j ect to a knapsac k constrain t ( k = 1). Previously , the b est b ound wa s O (log n ) [9]. Both algorithms pro vide the same appr o ximation ratio also for non-monotone subadditive v alu- ations, not just monotone ones. In the app endix w e sh o w that when the v aluatio n is non -mon otone and su b mo dular, then the inte gralit y gap is 2. Theorem 5.1 The fol lowing two algorithms exist: • A 2 -appr oximation algorithm for maximizing a (not ne c essarily monotone) sub add itive f u nc- tion subje ct to a c ar dinality c onstr aint that uses p olynomial ly many demand queries. • A (1 + k 1 − ǫ ) -appr oximation algorithm for maximizing a (not ne c essarily monotone) sub additive function subje ct to k -knapsack c onstr aints that uses p olynomial ly many demand queries, for every c onstant ǫ > 0 and c onstant k . W e start with th e first algorithm, wh ic h u ses a simple round ing sc heme. The Algorithm 1. Obtain a strict fractional solution x S 1 = α and x S 2 = 1 − α . 2. Arbitrarily divide S 2 in to sets U 1 , . . . , U l suc h that for eac h i < l , | U i | = k and | U l | ≤ k . 3. Outp u t A = arg max( v ( S 1 ) , v ( U 1 ) , . . . , v ( U l )). Notice that the ab o ve algorithm can b e implemen ted with a p olynomial num b er of d emand qu eries. Lemma 5.2 L et A b e the bund le that the algorithm outputs. Then, 2 v ( A ) ≥ x S 1 v ( S 1 ) + x S 2 v ( S 2 ) . Pro of: x S 1 v ( S 1 ) + x S 2 v ( S 2 ) ≤ x S 1 v ( S 1 ) + x S 2 · l · max( v ( U 1 ) , . . . , v ( U l )) ≤ ( x S 1 + x S 2 · l ) max ( v ( S 1 ) , v ( U 1 ) , . . . , v ( U l )) ≤ ( x S 1 + x S 2 ( k 2 k + 1)) v ( A ) ≤ (1 + x S 2 | S 2 | k ) v ( A ) ≤ (1 + 1) v ( A ) = 2 v ( A ) where the first inequality holds by sub additivit y v ( S 2 ) ≤ P l i =1 v ( U i ), the third uses l ≤ ⌈ k 2 k ⌉ , th e fourth inequalit y u ses x S 1 + x S 2 ≤ 1, and the last inequalit y h olds s in ce b y the L P w e hav e that x S 2 | S 2 | ≤ k . 11 5.1 An A ppro ximation Algorithm for Subadditiv e V aluations Sub ject to k - Knapsac k Const rain ts In this section we stu dy maximization sub ject to k knapsac k constraint s. In th is p roblem w e ha ve a v aluation v and k costs for eac h item j : c i 1 , . . . , c i k . Let C i ( S ) = Σ j ∈ S c i j . W e also hav e k b udgets B 1 , . . . , B k . Th e goal is to fin d a maximum-v alue bund le S such that for every i , C i ( S ) ≤ B i . W e note that e e +1 -appro ximation algorithms exist for maximizing a montone submo dular v al u - ation sub ject to k knapsac k constraints (the algorithms u s e only v alue queries) [5, 18] (note that our algorithms are for the more general subadd itiv e case). Con s ider a generalization of the L P for a single knapsac k constrain t: Maximize: P S x S · v ( S ) Subje ct to: • F or eac h constrain t i ∈ [ k ]: P S x S · C i ( S ) ≤ B i . • P S x S ≤ 1. • F or eac h bundle S : x S ≥ 0. Prop osition 5.3 The LP c an b e solve d with a p olynomial numb er of demand querie s. Pro of: Once again w e tak e its du al and give a s ep aration oracle to the d ual to solv e the LP . Minimize: P k i =1 y i · B i + z Subje ct to: • F or eac h bundle S : z + P k i =1 y i · C i ( S ) ≥ v ( S ). • F or eac h constrain t i ∈ [ k ]: y i ≥ 0. • z ≥ 0. Similar to the case of single kn apsac k constrain t (Prop osition 3.1) w e can solv e this dual LP by ellipsoid metho d. T he separation oracle for the dual is a demand qu ery m ax T v ( T ) − P k i =1 y i · C i ( T ). The fin al algorithm uses en umer ation o ve r “big” items. Definition 5.4 An i tem j is c al le d big if c i j ≥ ǫ · B i for some c onstr aint i ∈ [ k ] . An item i s c al le d small otherwise. Let B the set of b ig items and W b e the set of sm all items. The Algorithm 1. F or eac h set T ⊆ B suc h that for eac h constrain t i ∈ [ k ]: C i ( T ) ≤ B i : (a) Let M ′ = T ∪ W . (b) Obtain f ractional solution on items from M ′ . (c) Divide eac h bund le S in the f r actional s olution into sets, U S 1 , U S 2 , . . . , U S l S sets eac h satisfying bud get constraint eac h U S i resp ects all bu dget constr aints. Additionally l S ≤ ⌈ P k i =1 C i ( S ) B i (1 − ǫ ) ⌉ . 12 (d) Let A T = arg max( U S j ). 2. Outp u t arg max T ⊆ B v ( A T ). Notice that Step (1c) can b e implemented for eac h S as follo ws: pu t the su b set T ∩ S in U S 1 and add items from S without violating th e bud get constraint of U S 1 . No w d ivid e the rest of the items S in to b undles U S 1 , U S 2 , . . . , U S l S so that eac h su ch b undle resp ects the budget constrain t and for eac h i , B i − C i ( U S r ) ≤ ǫ · B i . Since we only ha v e to handle sm all items, we get that l S ≤ ⌈ P k i =1 C i ( S ) B i (1 − ǫ ) ⌉ . As for the n umb er of demand queries w e mak e, notice that in eac h set T we consider all items are big. Therefore, for this set to resp ect eac h of the bud get constrain ts it must b e that | T | ≤ k /ǫ , whic h implies that the num b er of iterations w e make is at most m k ǫ . Since in eac h iteration we mak e a p olynomial num b er of d emand queries, if ǫ is constan t the total num b er of qu eries w e mak e is indeed p olynomial. Lemma 5.5 L et A b e the bund le that the algorithm outputs. Then, (1 + k 1 − ǫ ) v ( A ) ≥ P S x S v ( S ) . Pro of: C onsider the s et A T that w as obtained in the iteration where T is exactly the set of big items in the optimal solution O . Th e optimal fraction solution with resp ect to W ∪ T has the same v alue as the un restricted fractional solution. W e can therefore analyze the approximat ion r atio for this A T . X S x S v ( S ) ≤ X S x S · l S · max( v ( U S 1 ) , v ( U S 2 ) , . . . , v ( U S l S )) ≤ v ( A T ) X S x S · l S ≤ v ( A T ) X S x S ( ⌈ k X i =1 C i ( S ) B i (1 − ǫ ) ⌉ ) ≤ v ( A T ) X S x S ( k X i =1 C i ( S ) B i (1 − ǫ ) + 1) ≤ v ( A T )(1 + X S x S k X i =1 C i ( S ) B i (1 − ǫ ) ) = v ( A T )(1 + k X i =1 1 B i (1 − ǫ ) X S x S C i ( S )) ≤ v ( A T )(1 + k X i =1 1 B i (1 − ǫ ) B i ) ≤ v ( A T )(1 + k 1 − ǫ ) where the second inequ alit y holds since b y su b additivit y v ( S ) ≤ Σ l S i =1 v ( U S i ), and the second to last inequalit y holds since P S x S C i ( S ) ≤ B i (b y the constrain ts of th e fractional solution). 13 6 Lo w er Bounds 6.1 A Tight Lo w er B ound for XOS V aluations W e show that our algorithms f rom Section 5 are essentia lly tigh t: an appro ximation ratio of 2 − ǫ requ ires exp onen tially many demand queries. W e n ote that provi n g b oun ds on the p o we r of algorithms with demand queries turned out to b e n ot an easy task, and in particular very different than showing lo w er b ounds on the p ow er of v alue queries. F or example, when we consider v alue queries there are only finitely many v alue qu eries that we ha v e to consider (no more than the num b er of su bsets of M ), whereas there is an infi nite num b er of demand queries we ha v e to consider. The crux of our pro of is to sho w that the p ow er of any arbitr ary d emand query is limited in some formal sense, hence exp onent ially man y demand queries are needed to d istinguish b et we en the case th at the optimal v alue is 2, and b et w een the case it is 1 + ǫ . W e note that that our b ound holds also for r andomized mec hanisms , and in fact h olds also for a the class of fractionally su badditive v aluations, a strict sub class of sub additiv e v aluations. Theorem 6.1 L et A b e a r andomize d algorithm that achieves a (2 − ǫ ) - appr oximation, for some fixe d ǫ > 0 . L et α b e suc h that m = k α . A makes in e xp e ctation at le ast 1 2 · k ǫk 100 ( α − 1) m ( 400 e ǫ 2 ) ǫk 100 (2 e ) k − ǫk 100 demand queries, even if the valuation is XOS. In particular, fix some ǫ, γ > 0, an d let k = m 1 − γ . The theorem sa ys that obtaining a 2 − ǫ appro ximation requires exp onen tial n umb er of demand quer ies. The follo wing thr ee families of additiv e v aluati ons will b e used in th e pro of: 1. F or ev ery item j , define the v aluation I j suc h that I j ( { j } ) = 1 an d for ev ery t 6 = j , I j ( { t } ) = 0. 2. F or every subset T , | T | = k , , d efine the v aluation G T suc h that G T ( { j } ) = 2 k if j ∈ T , an d for ev ery j / ∈ T , G T ( { j } ) = 0. 3. The v aluati on B suc h that for ev ery item j , B ( { j } ) = 1+ ǫ k . F or ev ery T , | T | = k , let v T ( S ) = max( I 1 ( S ) , . . . , I m ( S ) , B ( S ) , G T ( S )). In add ition, let v ∅ ( S ) = max( I 1 ( S ) , . . . , I m ( S ) , B ( S )). Notice that the v aluatio n s are XOS (i.e., fractionally su badditiv e) b y d efinition. W e w ill pr o v e that: Lemma 6.2 Determining whether the valuation is v ∅ (and not v T , for some T ) r e qu i r e s in exp e c- tation at le ast 1 2 · k ǫk 100 ( α − 1) m ( 400 e ǫ 2 ) ǫk 100 (2 e ) k − ǫk 100 demand queries. Lemma 6.2 implies T heorem 6.1: the bu ndle T is of size k and v T ( T ) = 2. On th e other h and, for ev ery bund le S of s ize k it holds th at v ∅ ( S ) = 1 + ǫ . Pro of: Sp ecifically , w e sh o w that an y deterministic algorithm must m ak e in exp ectation the sp ecified num b er of demand queries to determine if the v aluation is v ∅ , wh en the v aluat ion is c hosen uniformly at random from th e set { v ∅ } ∪ T : | T | = k v T . Y ao’s prin ciple deliev ers the lemma n o w. W e use the follo wing d efinition: Definition 6.3 Fix a demand qu e ry dq = ( p 1 , ..., p m ) . We say that dq co v ers v T if the demand set of dq in v T c onta ins a set S such that G T is the maximizing clause of S . 14 Claim 6.4 L et dq = ( p 1 , ..., p m ) b e a demand query, and let v T b e a valuation that is c over e d by dq . L et S ⊆ T b e some bund le in the demand set of dq in v T such that G T is the maximizing clause of S . Then, Σ j ∈ S p j ≤ 2 | S | k − 1 + 2 k . F urthermor e, we have that | S | ≥ k 2 − 1 . Pro of: Let t = arg min j ∈ S p j . G T (and not I t ) is the maximizing clause for S , and thus: G T ( S ) − Σ j ∈ S p j = 2 | S | k − Σ j ∈ S p j ≥ 1 − p t = I t ( t ) − p t Since S is in the demand set and G T is its maximizing clause, for eac h j ∈ S , p j ≤ 2 k (otherwise S − { j } is more profitable than S , and thus S is not in the demand set). 2 | S | k − Σ j ∈ S p j ≥ 1 − 2 k Reorganizing w e h av e that 2 | S | k − 1 + 2 k ≥ Σ j ∈ S p j , as n eeded. As for the s econd part of the claim, notice that if | S | < k 2 − 1 then v T ( S ) = G T ( S ) < 1 − 2 k , and therefore G T is not the m aximizing clause of S sin ce I T ( t ) − p t < 1 − 2 k . The k ey observ ation is that if v T is n ot cov ered b y dq , then the demand set of dq is identica l in v ∅ and v T . In particular, the only inform ation that executing dq adds is either to determine that the v aluation is some sp ecific v T (in case v T is co v ered by dq ) or to claim that the v aluation is not an y of the v T ’s that are co v ered b y dq . Ho we ver, to determine whether the v aluati on is v ∅ , we need to rule out al l p ossible v alues of T . The h eart of th e pro of is the follo wing claim: Claim 6.5 Fix a demand query dq = ( p 1 , ..., p m ) . The numb er of bund les that dq c overs is at most  4 k ǫ ǫk 100  ·  m − ǫk 100 k − ǫk 100  . Pro of: W e upp er b ound th e n umb er of v aluations V T that dq co vers b y describin g a set L of items of size at most | L | ≤ 4 k ǫ with the prop erty that for ev ery v T that is co v ered by dq , | L ∩ T | > ǫk 100 . The num b er of th ese bu ndles is at most: k X i = ǫk 100  4 k ǫ i  m − 4 k ǫ k − i  ≤  4 k ǫ ǫk 100  ·  m − ǫk 100 k − ǫk 100  W e no w describ e a pro cess that constru cts the set L . After that, we p r o v e that L is in d eed of the sp ecified size. 1. Let L 0 = ∅ , T = { T | v T is co vered by dq } , and i = 0. 2. While T 6 = ∅ : (a) Let i = i + 1. (b) Select some T i ∈ T . (c) Let S T i b e a s et in the d emand set of dq in v T that is m aximized in the clause G T . (d) Let L i = L i − 1 ∪ S T i . (e) Let T = { T | v T is co v ered b y dq and | L i ∩ T | < ǫk 100 } . 3. Let L = L i . 15 W e now show that | L | ≤ 4 k ǫ . Sp ecifically , w e sh o w that if the n umb er of iterations t is more than 4 ǫ then the p rofit of L is at least 2. This m eans the demand set do es n ot con tain bu n dles of size at most k at all, since their v alue is at most 2. Hence dq co v ers no v aluation v T (i.e., v ( L ) − Σ j ∈ L p j ≥ v ( T ) − Σ j ∈ T p j for every b undle T of size at most k ). Th us, if dq co vers some v aluati on v T it must hold that t ≤ 4 ǫ . F or eac h iteration i , let S i = L i − L i − 1 . Observe that b y Claim 6.4, | S T i | ≥ k 2 − 1 and therefore | S i | ≥ k 2 − 1 − ǫk 100 . F rom the same claim w e also ha ve that Σ j ∈ S T i p j ≤ 2 | S T i | k − 1 + 2 k . T ogether this implies th at Σ j ∈ S i p j ≤ 2( | S i | + k · ǫ 100 ) k − 1 + 2 k . W e would like to b oun d the n umb er of iteratio ns t th e pro cess go es on. By our discussion ab o ve, the p ro cess m ust stop b efore the profit of L is bigger than 2 (i.e., v ∅ ( L ) − Σ j p j > 2): v ∅ ( L ) − Σ j p j = (1 + ǫ ) | L | k − Σ j ∈ L p j = (1 + ǫ )Σ i ≤ t | S i | k − Σ i ≤ t Σ j ∈ S i p j ≥ (1 + ǫ )Σ i ≤ t | S i | k − Σ i ≤ t ( 2 | S i | k − 1 + 2 k + ǫ 100 ) = t · (1 − ǫ 100 − 2 k ) − (1 − ǫ )Σ i ≤ t | S i | k ≥ t · (1 − ǫ 100 − 2 k ) − (1 − ǫ ) · t = t · ( ǫ − ǫ 100 − 2 k ) ≥ t · ǫ 2 where the second to last in equ alit y follo ws since for eac h i , | S i | ≤ | T i | ≤ k . Th erefore for the profit of L to b e at most 2 it m u s t hold that t ≤ 4 ǫ . Recall that an y iteration we are adding to L at most k items. Therefore | L | ≤ t · k ≤ 4 k ǫ . Claim 6.5 implies Lemma 6.2: the total num b er of bun dles of size k is  m k  . T o rule out at least ( m k ) 2 p ossible bundles, the n umb er of demand queries d we ha v e to mak e is at least (w e use the b ound s  n r  ≥ ( n r ) r ,  n r  ≤ ( ne r ) r and us e m = k α ): 1 2 ·  k α k   4 k ǫ ǫk 100  ·  k α − ǫ · k 100 k − ǫk 100  ≥ 1 2 · k αk k k ( e 4 k ǫ ǫk 100 ) ǫk 100 · ( e k α − ǫ · k 100 k − ǫk 100 ) k − ǫk 100 = 1 2 · k k ( α − 1) · ( k − ǫk 100 ) k − ǫk 100 ( 400 e ǫ 2 ) ǫk 100 ( ek α ) k − ǫk 100 ≥ 1 2 · k k ( α − 1) · k k − ǫk 100 ( 400 e ǫ 2 ) ǫk 100 (2 ek α ) k − ǫk 100 = 1 2 · k k · α − ǫ · k 100 ( 400 e ǫ 2 ) ǫk 100 (2 e ) k − ǫk 100 · k α · k − αǫk 100 ≥ 1 2 · k ǫk 100 ( α − 1) ( 400 e ǫ 2 ) ǫk 100 (2 e ) k − ǫk 100 16 Therefore, u n til we r u le out at least half of the bundles, an y additional d emand query finds T with probabilit y at m ost 1 d . T o rule out h alf of th e bund les w e hav e to mak e at least d queries, hence after d m queries we find T with probab ility at most o (1). 6.2 Ruling Out an FPT AS for Submodular Maximization Sub ject to a Knap- sac k Constraint W e show that there is no FPT AS for the prob lem of op timizing a submo d lar fu nction sub ject to a knapsac k constrain t (an FPT AS is a (1 + ǫ )-appro ximation algorithm that the n u mb er of demand queries it mak es is p oly ( m, 1 ǫ )). Theorem 6.6 L et A b e a r andom ize d algorithm that achieves an m 2 − 1 3 m 2 − 1 2 -appr oximation. A makes in exp e ctation at le ast 2 m ·  m 2 m 4  demand q ueries. Notice that th e lo w er b ound on the appr o ximation ratio dep end s p olynomially on m and there- fore an FPT AS must ac hieve a b etter ratio in time pol y ( m, 1 ǫ ). This sh o ws that an FPT AS for this problem is imp ossible. F or pro of, we consider the follo wing family of instances. Let A b e some set of m 2 items, and let B b e the set th at con tains the rest of the items. F or eac h item j ∈ A let c j = 2 m − ǫ , and for eac h item j ∈ B let c j = 2 m + ǫ . Let the total b u dget b e 1. F or ev ery T of size n 2 suc h that | T ∩ A | = m 4 define the follo wing v aluations: v T ( S ) =            | S | , | S | < m 2 ; m 2 , | S | > m 2 ; m 2 , | S | = m 2 , C ( S ) > 1; m 2 − 1 3 , | S | = T ; m 2 − 1 2 , otherwise. v ∅ ( S ) =        | S | , | S | < m 2 ; m 2 , | S | > m 2 ; m 2 , | S | = m 2 , C ( S ) > 1; m 2 − 1 2 , otherwise. Lemma 6.7 Determining whether the valuation is v ∅ (and not v T , for some T ) r e qu i r e s in exp e c- tation at le ast 2 m ·  m 2 m 4  demand queries. Lemma 6.7 implies T heorem 6.6: if the v aluation is v T then v T ( T ) = m 2 − 1 3 . Otherwise, for ev ery bund le other S of b elo w the b udget it h olds that v ∅ ( S ) ≤ m 2 − 1 2 . Pro of: W e us e the follo wing definition: Definition 6.8 Fix a demand qu e ry dq = ( p 1 , ..., p m ) . We say that dq co v ers v T if the demand set of dq in v T c onta ins T . Claim 6.9 L et dq = ( p 1 , ..., p m ) b e a demand query, and let v T , v T ′ b e two valuations that ar e c over e d by dq . Then e i ther T ∩ A = T ′ ∩ A or T ∩ B = T ′ ∩ B . Pro of: Supp ose not. Let a, b ∈ T and a ′ , b ′ ∈ T ′ b e su c h that a, b / ∈ T ′ , a ′ , b ′ / ∈ T , a, a ′ ∈ A , b, b ′ ∈ B . Defin e the follo wing bun dles: T b ′ = T − a + b ′ , T ′ a = T ′ − b ′ + a 17 If the v aluation is v T , then T is in the demand set and therefore v T ( T ) − p ( T ) ≥ v T ( T b ′ ) − p ( T b ′ ). Since C ( T b ) > 1, we ha v e that 1 3 ≤ p b ′ − p a . If th e v aluation is v T ′ , then we hav e that v T ′ ( T ′ ) − p ( T ′ ) ≥ v T ′ ( T ′ a ) − p ( T ′ a ). Since C ( T ′ a ) < 1 w e ha ve that 1 6 ≥ p b ′ − p a . W e hav e r eac h ed a contradictio n and the claim follo ws. Observe that if v T is not co v ered by dq , then the demand set of dq is iden tical in v ∅ and v T . In particular, the only information that executing dq add s is either that the v aluatio n is some sp ecific v T (in case v T is co v ered b y dq ) or provide a pro of that the v aluation is not any of the v T ’s th at are co v ered by dq . How ev er, to determine whether the v aluatio n is v ∅ , w e need to rule out al l p ossible v alues of T . S upp ose that we ha ve d demand qu eries that fail. T here are  m 2 m 4  ·  m 2 m 4  p ossible v alue of T (half of the items in T come from A an d the other h alf from B ), and by the claim eac h demand query can r u le out at most  m 2 m 4  (w e keep all items in, sa y , A , and c ho ose m 4 items from B ). Therefore, u n til we r u le out at least half of the bundles, an y additional d emand query finds T with probabilit y at most 1 2 ( m 2 m 4 ) . T o rule out h alf of the bu ndles we therefore ha v e to mak e in exp ectation at least  m 2 m 4  queries, hence after 2 m ·  m 2 m 4  queries we find T with probab ility at most o (1). Ac kno wledgmen ts W e thank Bobb y Klein b erg for helpful discuss ions . References [1] Maria-Florina Balcan, Avrim Blum, and Yisha y Mansour . Item pricing for r ev en ue maximiza- tion. In EC’08. [2] Y air Bartal, Rica Gonen, and Noam Nisan. I n cen tiv e compatible m ulti unit com binatorial auctions. In T ARK’03. [3] Liad Blumrosen and Noam Nisan. 2007. Com binatorial Auctions (a sur vey). I n “Algorithmic Game Theory”, N. Nisan, T. Roughgarden, E. T ardos and V. V azirani, editors. [4] Liad Blumr osen and Noam Nisan. On the computational p ow er of demand queries. SIAM J. Comput. , 39(4):1372 –1391, 2009. [5] C h andra Chekur i, Jan V ondr´ ak, and Rico Zenklu sen. Submo d ular function maximization v ia the multi linear r elaxatio n and cont entio n resolution schemes. In STO C’11. [6] S hahar Dobzinski. An imp ossibilit y result for tr uthful com binatorial auctions with su bmo du lar v aluati ons. In S TOC’11. [7] S hahar Dobzinski, Noam Nisan, and Mic hael Schapira. Ap pro ximation algorithms for com bi- natorial auctions with complement-free bidd ers. In ST OC’05. [8] S hahar Dobzinski, Noam Nisan, and Mic hael Sc hapira. T ruthf ul randomized mec hanism s for com binatorial auctions. In STOC’06. [9] S hahar Dobzinski, Ch ristos Pa padimitriou, an d Y aron S inger. Mec hanisms for complement- free pro cu remen t. In EC’11. 18 [10] Sh ahar Dobzinski and Michae l S c hapira. An impr o v ed ap p ro ximation algorithm for combina- torial auctions with s u bmo du lar b idders. In S OD A’06. [11] Uriel F eige. On m aximizing welfare where the utilit y fun ctions are su badditiv e. In STOC ’06. [12] Uriel F eige. A th r eshold of ln n f or approximat ing set co v er. Journal of the ACM , 45(4):63 4– 652, 1998. [13] Uriel F eige and Jan V ond r´ ak. App ro ximation algorithms for allo cation pr oblems: Improving the factor of 1-1/e. In FOCS’06. [14] M. L. Fisher, G. L. Nemhauser, and L. A. W olsey . An analysis of appro ximations for maxi- mizing su b mo dular set f u nctions - ii. In Polyhe dr al Combinatorics , v olume 8 of Mathematic al Pr o gr amming Studies , pages 73–87. S p ringer Berlin Heidelb erg, 1978. [15] Sh a y an Oveis Gharan and Jan V ondr´ ak. S ubmo d u lar maximization by simulated annealing. In SO D A’11. [16] Su bhash Khot, Ric hard J. Lipton, Ev angelo s Mark akis, and Aran ya k Meh ta. In appro ximability results for com binatorial auctions with su bmo du lar utility fun ctions. In WINE’05. [17] Ariel Ku lik, Hadas S hac hnai, and T ami T amir. Maximizing sub mo dular set f u nctions sub ject to multiple linear constraints. In SODA’0 9. [18] Ariel Kulik, Hadas Shac hn ai, and T ami T amir. Appro ximations for monotone and non- monotone sub mo dular maximization w ith kn apsac k constraint s. CoRR , abs/1101.29 40, 2011. [19] Se’bastien Lahaie, Florin C onstan tin, and David C. P arkes. More on the p o w er of demand queries in com binatorial auctions: Learning atomic languages and handling incenti ves. In IJCAI’05. [20] Ron La vi and Chaitan ya Swam y . T ruth f ul and n ear-optimal m ec hanism design via linear programming. In F OCS’05. [21] Jon Lee, V ahab S. Mirrokni, Visw anath Nagara jan, and Maxim S v ir idenk o. Non-monotone submo d ular maximizati on under matroid and kn apsac k constraint s. In STOC ’09. [22] Benny Lehm an n , Daniel Lehmann, and Noam Nisan. Combinato rial auctions with decreasing marginal utilities. In EC ’01. [23] G. L. Nemhauser and L. A. W olsey . Best algorithms for app ro ximating the maxim um of a submo d ular set function. Mathematics of Op er atio ns R ese ar ch , 3(3):177–18 8. [24] Noam Nisan and Ilya Segal. Exp onential comm unication inefficiency of d emand qu er ies. In T ARK’05. [25] T uomas Sand h olm and Craig Boutilier. 2006. Preference Elicitation in Com binatorial Auctions. In “Combinatoria l Au ctions”, P . Cr am ton and Y. S hoham and R. Steinberg, editors. [26] Y aron Singer. Budget feasible mec han ism s. In F OCS ’10. [27] Jan V ondr´ ak. Optimal approxima tion f or the submo dular welfare p roblem in the v alue oracle mo del. In STOC’08. 19 A App endix for Section 4 Pro of of Claim 4.4 The pr o of is b asically an algebraic manipu lation: γ = max k 1 ≤ k ≤ k 2 α · k 2 k 1 + k 2 − k + ((1 − α ) − α · k − k 1 k 1 + k 2 − k ) k 2 k = max k 1 ≤ k ≤ k 2 k 2 − k k 2 − k 1 · k 2 k 1 + k 2 − k + ( k − k 1 k 2 − k 1 − k 2 − k k 2 − k 1 · k − k 1 k 1 + k 2 − k ) k 2 k = max k 1 ≤ k ≤ k 2 k 2 − k k 2 − k 1 · k 2 k 1 + k 2 − k · (1 − k − k 1 k ) + k − k 1 k 2 − k 1 · k 2 k = max k 1 ≤ k ≤ k 2 k 2 − k k 2 − k 1 · k 2 k 1 + k 2 − k · k 1 k + k − k 1 k 2 − k 1 · k 2 k = max k 1 ≤ k ≤ k 2 k 2 k ( k 2 − k 1 ) · k ( k 1 + k 2 ) − k 2 − k 2 1 k 1 + k 2 − k = max k 1 ≤ k ≤ k 2 k 2 ( k 2 − k 1 ) · (1 − k 2 1 k ( k 1 + k 2 − k ) ) = max k 1 ≤ k 2 k 2 ( k 2 − k 1 ) · (1 − 4 k 2 1 ( k 1 + k 2 ) 2 ) (3) = max k 1 ≤ k 2 k 2 ( k 2 − k 1 ) · k 2 2 + 2 k 1 k 2 − 3 k 2 1 ( k 1 + k 2 ) 2 = max k 1 ≤ k 2 k 2 ( k 2 − k 1 ) · ( k 2 + 3 k 1 ) · ( k 2 − k 1 ) ( k 1 + k 2 ) 2 = max k 1 ≤ k 2 ( k 2 + 3 k 1 ) · k 2 ( k 1 + k 2 ) 2 = max x ≥ 1 ( x + 3) · x ( x + 1) 2 = 9 8 where in (3) we us e the fact that the expression is maximized at k = k 1 + k 2 2 . In the second to last equation we set x = k 2 k 1 . B In tegralit y Gaps An Integralit y Gap of 9 8 for Monotone Submo dular V aluations Theorem B.1 Ther e e xists a monotone submo dular valuation v for which the inte gr ality gap for maximization sub j e ct to a c ar dinality c onstr aint is 9 8 . Pro of: T o wa rd s defining the v aluatio n v , consider a six-elemen t universe U = { e 1 , . . . , e 6 } . Define the follo wing four sets: m 1 = { e 1 , e 2 , e 3 } , m 2 = { e 1 , e 4 } , m 3 = { e 2 , e 5 } , m 4 = { e 3 , e 5 } . No w define the follo wing v aluation wh ere the set of item is M = { m 1 , m 2 , m 3 , m 4 } . Let v b e the follo wing v aluati on: v ( S ) = | ∪ m i ∈ S m i | , i.e., the num b er of elemen ts in U that the union of the sets in S co v ers. This v aluatio n is sub mo dular. No w let th e cardinalit y constrain t b e k = 2. The optimal solution is to tak e any set S of size 2. The v alue of the optimal integ ral solution is 4. On the other h and, consider the follo wing fr actional 20 solution: x m 1 = 1 2 and x m 2 ,m 3 ,m 4 = 1 2 . T he f r actional s olution resp ects the cardinalit y constrain t and has a v alue of 4 . 5. The in tegralit y gap is th er efore 9 8 . An Integralit y Gap of 2 for Non-Monotone Submo dular V a lut ions Theorem B.2 F or every c onstant ǫ > 0 , ther e exists a non-monotone submo dular valuation v for which the inte gr ality gap for maximization subje ct to a c ar dinality c onstr aint is 2 − ǫ . Pro of: Let the items b e { 1 , . . . , k 2 + 1 } . Define the follo wing non-monotone su bmo du lar v aluation: v ( S ) = ( 1 − | S | k 2 , S ⊆ { 2 , 3 , . . . , k 2 + 1 } , 1 ∈ S ; | S | k , S ⊆ { 2 , 3 , . . . , k 2 + 1 } . Then the optimal in tegral solution has a v alue of 1, whereas the fractional solution x { 1 } = k k +1 , x { 2 , 3 ,...,k 2 +1 } = 1 k has v alue of 1 · k k +1 + k 2 · 1 k +1 = 2 k k +1 . As k tend s to in finit y the in tegralit y gap tends to 2. 21