Approximating Multi-Criteria Max-TSP

Appro ximating Multi-Criteria Max-TSP ∗ Markus Bl¨ aser Bo do Man they Oliver Putz Saarland Univ er sit y , Computer Science Postfac h 151150 , 66041 Saarbr ¨ uck en, German y blaese r/manthe y@cs.uni-sb.de , oli.pu tz@gmx.d e W e presen t randomized appro ximation algorithms for multi-criteria Max-TSP . F or Max-STSP with k > 1 ob jectiv e f u nctions, w e obtain an approxi mation r ati o of 1 k − ε f or arbitrarily smal l ε > 0. F or Max-A TSP with k ob jectiv e fun ctio ns , we obtain an appro ximation ratio of 1 k +1 − ε . 1 Multi-Criteria T ra v eling Salesman Problem 1.1 T rav eling Salesm an Problem The tra veling salesman problem (TS P) is one of the most fun damen tal problems in com bina- torial optimiza tion. Giv en a graph, the g oal is to find a Hamiltonian cyc le of minim um or maxim um weig ht. W e consider fi n ding Hamiltonian cycles of maximum w eigh t (Max-TSP). An instance of Max-TSP is a complete graph G = ( V , E ) with edge w eights w : E → N . Th e goal is to fin d a Hamiltonian c ycle of maxim u m weigh t. The we ight of a Hamiltonian cycle (or, more general, of a subset of E ) is the su m of the we ights of its edges. If G is un directed, w e sp eak of Max-STSP (symmetric TSP). If G is directed, w e ha v e Max-A TS P (asymmetric TSP). Both Max-STSP and Max-A TSP are NP-hard a nd APX-hard. Th u s, w e are in n eed of appro ximation algorithms. The currently b est ap p ro ximation algorithms for Max-STSP and Max-A TS P achiev e appr o ximation ratios of 61 / 81 and 2 / 3, resp ectiv ely [2, 5]. Cycle co vers are an imp ortan t to ol for designing appro ximation algorithms for the TSP . A cycle co v er of a graph is a set of v ertex-disjoin t cycles suc h that eve r y vertex is part of exactly one cycle. Hamiltonian cycles are s p ecial cases of cycle co v ers that consist of j ust one cycle. Thus, the weig ht of a maximum-w eigh t cycle cov er is an upp er b oun d for the wei ght of a m aximum-w eigh t Hamilto n ia n cycle. In con trast to Hamiltonian cycles, cycle co vers of minim um or maximum weigh t can b e computed efficien tly us ing matc hing algorithms [1]. 1.2 Multi-Criteria Opt imiza tion In many optimization p r oblems, there is more than one ob jectiv e f unction. Consider b uying a car: W e migh t wa nt to bu y a c heap, fast car with a go o d gas mileage. Ho w do we decide ∗ An exten ded abstract of this work will app ear in Pr o c. of the 16th Ann. Eur op e an Symp osium on Algor i th ms (ESA 2008) . 1 whic h car suits u s b est? With m ultiple criteria inv olv ed, there is no natural notion of a b est c hoice. Ins te ad, we ha ve to b e conte nt with a trade-o ff. The aim of m u lti- criteria optimization is to cop e with this problem. T o transfer the concept of an optimal solution to m u lti-criteria optimization problems, the notion of Par eto c u rves wa s introd uced (c f. Eh rgott [ 3]). A Pareto curv e is a set of solutions that can b e considered optimal. More formally , a k -crite ria optimization problem consists of instances I , solutions sol( X ) for ev ery ins ta nce X ∈ I , and k ob jectiv e functions w 1 , . . . , w k that map X ∈ I and Y ∈ sol( X ) to N . Throughout this pap er, our aim is to maximize the ob jectiv e fu nctions. W e sa y that a solution Y ∈ sol( X ) dominates another solution Z ∈ sol( X ) if w i ( Y , X ) ≥ w i ( Z, X ) for all i ∈ [ k ] = { 1 , . . . , k } and w i ( Y , X ) > w i ( Z, X ) for at least one i . T h is means that Y is str ict ly preferable to Z . A Par eto curve (also known as Par eto set or efficient set ) for an ins ta n ce con tains all solutions of that in s ta n ce th at are not d ominate d by another s ol u tio n. Unfortunately , Paret o curves cannot b e computed efficien tly in many cases: First, they are often of exp onen tial size. Second, b ecause of straigh tforward reductions from knapsac k problems, they are NP-hard to compute ev en for otherwise easy pr oblems. Thus, we ha v e to b e conte nt with approxima te P areto cur ves. F or simpler notation, let w ( Y , X ) = ( w 1 ( Y , X ) , . . . , w k ( Y , X )). W e will omit the instance X if it is clear from the context. Inequalities are meant comp onen t-wise. A set P ⊆ sol( X ) of solutions is called an α appr oximate Par eto curve for X ∈ I if the follo win g holds: F or every solution Z ∈ sol( X ), th ere exists a Y ∈ P with w ( Y ) ≥ αw ( Z ). W e ha ve α ≤ 1, an d a 1 appro ximate Pareto curve is a Pa reto cu r v e. (This is n ot precisely true if there are sev eral solutions whose ob jectiv e v alues agree. How ever, in our case this is inconsequentia l, and we will not elab orate on th is for the sake of clarity .) An algorithm is called an α appr oximation algorithm if, give n the instance X , it computes an α appr o ximate Pareto curve. It is called a randomized α ap p ro ximation algorithm if its su cc ess probability is at least 1 / 2. This su cc ess probabilit y can b e amplified to 1 − 2 − m b y executing the algorithm m times and taking the union of all sets of solutions. (W e c an also remo ve solutions from this union that are dominated b y other solutions in the union, but this is not r equired b y the d efinition of an approximat e P areto cur ve.) P apadimitriou and Y annak akis [10] s ho wed that (1 − ε ) app ro ximate Pa r eto curves of size p olynomial in the instance size and 1 /ε exist. The tec hnical requirement f or the existence is that the ob jectiv e v alues of solutions in sol( X ) are b ound ed from ab o v e by 2 p ( N ) for some p olynomial p , wh ere N is the size of X . This is fulfilled in most natural optimization problems and in particular in our case. A f ul ly p olynomial time appr oximation scheme (FPT AS) for a multi-crite ria optimizatio n problem computes (1 − ε ) appro ximate P areto curve s in time p olynomial in the size of the instance and 1 /ε for all ε > 0. P apadimitriou and Y ann ak akis [10], based on a resu lt of Mulm uley et al. [9], show ed that multi-c riteria minimum-w eigh t matc hing admits a r andomize d FPT AS , i. e., th e algorithm succeeds in computing a (1 − ε ) approximate P areto curve with constan t pr obabilit y . This randomized FPT AS yields also a randomized FPT AS for the m ulti- criteria maxim um -w eight cycle co v er problem [8], whic h we will use in th e follo wing. Man they and Ram [6, 8] designed randomized app ro xim ation algorithms for sev eral v ariants of m ulti-criteria Min-TS P . Ho wev er, they lea ve it as an op en problem to design an y app r o xi- mation algorithm for Max-TSP . 2 1.3 New Results W e d evise the first appr o ximation algorithm for m ulti-criteria Max-TSP . F or k -criteria Max- STSP , w e ac hiev e an appro ximation ratio of 1 k − ε for arbitrarily small ε > 0. F or k -criteria Max-A TS P , we ac hiev e 1 k +1 − ε . Our algorithm is randomized. Its r u nning-time is p olynomial in the inpu t size and 1 /ε and exp onentia l in the n umb er k of criteria. Ho wev er, the num b er of differen t ob jectiv e functions is usually a small constan t. The main ingredient for our algorithm is a decomp osition tec h nique for cycle co vers and a reduction from k -criteria instances to ( k − 1)-criteria instances. 2 Outline and Idea A straigh t-forward 1 / 2 appr o ximation for mono-criterion Max-A TSP is the follo wing: First, w e compute a maxim um-weig ht cycle c ov er C . Then w e remo ve t he ligh test edge of eac h cycle, th us losing at most half of C ’s w eigh t. In this w a y , w e obtain a collection of paths. Finally , w e add edges to connect th e paths to g et a Hamiltonian cycle. F or Max-STSP , t he same approac h yields a 2 / 3 appro ximation since th e length of eve ry cycle is at least three. Unfortunately , th is do es not ge n erali ze to m ulti-criteria Max-TSP for which “ligh test edge” is usually not we ll defi n ed: If w e break an edge that has little weigh t w ith resp ect to one ob jectiv e, w e might lo se a lo t of w eigh t with resp ect to another ob jectiv e. Based on this observ ation, the basic idea b ehind our algorithm and its analysis is the follo wing case distinction: Light-weight e dges: If all ed ge s of our cycle cov er con tribu te only little to its weig ht, then remo ving one edge does not decrease the o v erall w eigh t by to o m uc h. No w we choose the edges to b e r emo ved su c h that no ob jectiv e loses to o m uc h of its w eigh t. He avy-weig ht e dges: If there is one edge that is v ery h ea vy with resp ect to at least one ob jectiv e, then we tak e on ly this edge from the cycle co ver. In this wa y , we ha ve enough we ight for one ob jectiv e, and we p roceed recursiv ely on the remaining graph with k − 1 ob jectiv es. In this wa y , the appro ximation ratio for k -criteria Max-TSP d epend s on t wo questions: First, how we ll can we decomp ose a cycle co ver consisting solely of ligh t-w eigh t edges? Second, ho w w ell can ( k − 1)-criteria Max-TSP b e approxima ted? W e deal with the fir s t question in Section 3. In Section 4, we present and analyze our appr o ximation algorithms, w hic h also giv es an answ er to the second question. Finally , w e giv e evidence that the analysis of the appro ximation ratios is tigh t and p oint ou t some ideas that migh t lead to b etter appr o ximation ratios (Section 5). 3 Decomp ositions Let α ∈ (0 , 1], and let C b e a cycle co ver. W e call a collection P ⊆ C of p aths an α - de c omp osition of C if w ( P ) ≥ αw ( C ). (Rememb er that all inequalities are mean t comp onent- wise.) In the follo wing, our aim is to fin d α -decomp ositions of cycle co v ers consisting solely of ligh t-w eigh t edges, th at is, w ( e ) ≤ αw ( C ) for all e ∈ C . Of course, not ev ery cycle co ver p ossesses an α -decomp osition for eve ry α . F or instance, a single dir ec ted cycle of length t w o, where eac h edge h as a w eigh t of 1 shows that α = 1 / 2 is b est p ossible for a single ob jectiv e fun ctio n in directed graphs . On the other hand, by remo ving the ligh test ed ge of eve ry cycle, we obtain a 1 / 2-decomp ositi on. 3 F or u ndirected graphs and k = 1, α = 2 / 3 is optimal: W e can find a 2 / 3-dec omp osition by remo ving the ligh test edge of ev ery cycle, and a single cycle of length three, where eac h edge w eigh t is 1, sho ws that this is tight. More general, we define α d k ∈ (0 , 1] to b e the maxim um num b er su ch that ev ery directed cycle co ve r C with w ( e ) ≤ α d k · w ( C ) for all e ∈ C p ossesses an α d k -decomp ositio n . Analogously , α u k ∈ (0 , 1] is the maximum num b er s uc h that ev ery un directed cycle co ve r C with w ( e ) ≤ α u k · w ( C ) p ossesses an α u k -decomp ositio n . W e ha ve α d 1 = 1 2 and α u 1 = 2 3 , as we ha ve already argued ab o ve. W e also hav e α u k ≥ α d k and α u k ≤ α u k − 1 as w ell as α d k ≤ α d k − 1 . 3.1 Existence of Decomp ositions In this section, we inv estigate for whic h v alues of α suc h α -decomp ositio n s exist. In th e subsequent s ec tion, w e show h o w to actually find go od decomp ositions. W e h av e already d ealt with α u 1 and α d 1 . Thus, k ≥ 2 remains to b e consider ed in the follo win g th eo rems. In particular, only k ≥ 2 is needed for the analysis of our algorithms. Let us first norm alize our cycle co vers to mak e the pr oofs in the follo wing a bit easier. F or directed cycle cov ers C , we can restrict ourselves to cycles of length t wo: If we ha ve a cycle c of length ℓ with edges e 1 , . . . , e ℓ , w e r eplac e it by ⌊ ℓ/ 2 ⌋ cycles ( e 2 j − 1 , e 2 j ) for j = 1 , . . . , ⌊ ℓ/ 2 ⌋ . If ℓ is o dd, then we add a edge e ℓ +1 with w ( e ℓ +1 ) = 0 and add the cycle ( e ℓ , e ℓ +1 ). (Strictly sp eaking, edges are 2-tuples of vertic es, and we cannot simply r eco nn ect them. What we mean is that we remo v e the edges of the cycle and create new edges with the same names and weigh ts together with app r opriate new vertic es.) W e do this f or all cycles of length at least thr ee and call the resu lting cycle co ver C ′ . Now any α -decomp osition P ′ of the new cycle co ve r C ′ yields an α -decomp osition P of th e original cycle cov er C by remo ving the newly added edges e ℓ +1 : In C , w e ha ve to remov e at least one edge of the cycle c to obtain a decomp osition. I n C ′ , we ha ve to remo ve at least ⌊ ℓ/ 2 ⌋ edges of c , thus at least one. F urtherm ore, if w ( e ) ≤ α · w ( C ) for ev ery e ∈ C , then also w ( e ) ≤ α · w ( C ′ ) for ev ery e ∈ C ′ since we k ept all edge w eigh ts. This also sho ws w ( P ) = w ( P ′ ). W e are inte rested in α -decomp ositions th at w ork for all cycle cov ers with k ob jectiv e f u nc- tions. Thus in p artic u la r , w e ha ve to b e able to decomp ose C ′ . T he consequence is that if eve ry directed cycle co v er that consists solely of cycles of length t w o p ossesses an α -decomp osition, then ev ery dir ec ted cycle co v er do es so. F or un directed cycle co v ers, we can restrict our selv es to cycles of length th r ee: W e replace a cycle c = ( e 1 , . . . , e ℓ ) b y ⌊ ℓ/ 3 ⌋ cycles ( e 3 j − 2 , e 3 j − 1 , e 3 j ) for 1 ≤ j ≤ ⌊ ℓ/ 3 ⌋ . If ℓ is not divisible by three, then w e add one or tw o edges e ℓ +1 , e ℓ +2 to f orm a cycle of length three with th e remaining edge(s). Again, ev ery α -decomp ositio n of th e new cycle co ver yields an α -decomp osition of the original cycle co v er. In the remainder of this section, we assu me that all d irect ed cycle co v ers consist solely of cycles of length t wo and all u n directed cycle cov ers consist solely of cycles of length three. Both theorems are prov ed u s ing the pr obabilisti c m etho d. 3.1.1 Undirected Cycle C o vers F or th e pro of of Th eorem 3.2 b elo w, w e u se Hoeffdin g’s inequalit y [4, T heorem 2], w hic h w e state here in a sligh tly mo dified ve r s io n . 4 Lemma 3.1 (Ho effding’s inequalit y) . L et X 1 , . . . , X n b e indep endent r andom variables, wher e X j assumes values in [ a j , b j ] . L et X = P n j =1 X j . Then P  X < E ( X ) − t  ≤ exp − 2 t 2 P n j =1 ( b j − a j ) 2 ! . Theorem 3.2. F or al l k ≥ 2 , we have α u k ≥ 1 k . Pr o of. Let C b e an y cycle co v er and w 1 , . . . , w k b e k ob jectiv e functions. First, w e scale the edge weig ht su c h that w i ( C ) = k for all i . Thus, w i ( e ) ≤ 1 for all edges e of C since the w eigh t of an y edge is at most a 1 /k fraction of the total w eight . Second, we can assu me that C consists solely of cycles of length three. Let c 1 , . . . , c m b e the cycles of C and let e 1 j , e 2 j , e 3 j b e the three edges of c j . W e p erform the follo wing random exp eriment: W e remov e one edge of eve r y cycle indep endently and uniformly at random to obtain a decomp osition P . Fix an y i ∈ [ k ]. Let X j b e the weigh t with resp ect to w i of th e path in P that consists of the t wo edges of c j . Then E ( X j ) = 2 w i ( c j ) / 3. Let X = P m j =1 X j . Then E ( w i ( X )) = 2 w i ( C ) / 3 = 2 k / 3. Ev ery X j assumes v alues b et we en a j = min { w i ( e 1 j ) + w i ( e 2 j ) , w i ( e 1 j ) + w i ( e 3 j ) , w i ( e 2 j ) + w i ( e 3 j ) } and b j = max { w i ( e 1 j ) + w i ( e 2 j ) , w i ( e 1 j ) + w i ( e 3 j ) , w i ( e 2 j ) + w i ( e 3 j ) } . Since the weig ht of eac h edge is at most 1, w e hav e b j − a j ≤ 1. S ince the sum of all edge weigh ts is k , we ha v e k ≥ m X j =1 b j ≥ m X j =1 b j − a j ≥ m X j =1 ( b j − a j ) 2 . Let us estimate the p robabilit y of the ev ent that X < 1, w hic h corresp onds to w i ( P ) < 1. If P ( X < 1) < 1 /k , then, by a union b oun d, w e ha ve P ( ∃ i : w i ( P ) < 1) < 1. Th us, P ( ∀ i : w i ( P ) ≥ 1) > 0, whic h implies th e existence of a 1 /k -decomp ositi on. By Ho effding’s inequalit y , P ( X < 1) = P  X < 2 k 3 −  2 k 3 − 1  ≤ exp − 2( 2 k 3 − 1) 2 k ! =: p k . W e ha v e p 4 ≈ 0 . 2494, p 5 ≈ 0 . 11, and p 6 ≈ 0 . 05. Thus, for k = 4 , 5 , 6, and also for all larger v alues of k , w e ha ve p k < 1 /k , whic h implies the existence of a 1 /k -d ec omp osition for k ≥ 4. The cases k = 2 and k = 3 remain to b e considered since p 3 ≈ 0 . 51 > 1 / 3 and p 2 ≈ 0 . 89 > 1 / 2. The b ound for α u 2 follo ws f rom Lemma 3.3 b elo w, whic h d oes n ot require w i ( e ) ≤ α u 2 · w i ( C ). Let us show α u 3 ≥ 1 / 3. This is done in a constructive wa y . First, we c h oose from ev ery cycle c j the edge e ℓ j that maximizes w 3 and p u t it into P ′ . The set P ′ will b ecome a sub set of P . Then w 3 ( P ′ ) ≥ 1. But w e can also ha ve some weigh t w ith resp ect to w 1 or w 2 . Let δ 1 = w 1 ( P ′ ) and δ 2 = w 2 ( P ′ ). If δ i ≥ 1, then w i do es not n eed an y further atten tion. Let C ′ = C \ P ′ . W e ha ve w i ( C ′ ) = 3 − δ i for i = 1 , 2, and C ′ consists solely of paths of length t wo. Of ev ery suc h path, we can c ho ose at most one edge for inclusion in P . (Cho osing b oth w ould create a cycle.) Let e 1 j , e 2 j b e the tw o edges of c j with w 2 ( e 2 j ) ≥ w 2 ( e 1 j ). No w we pro ceed by considerin g only w 2 . Let Q, Q ′ b e in itially empt y sets. F or all j = 1 , . . . , m , if w 2 ( Q ) ≥ w 2 ( Q ′ ), then we put (the hea vier edge) e 2 j in to Q ′ and e 1 j in to Q . I f w 2 ( Q ) ≤ w 2 ( Q ′ ), then w e pu t e 2 j in to Q and e 1 j in to Q ′ . Both P ′ ∪ Q and P ′ ∪ Q ′ are decomp ositions of C . W e claim that at least one has a w eigh t of at least 1 with resp ect to all three ob jectiv es. Since w 3 ( P ′ ) ≥ 1, this holds for b oth 5 with resp ect to w 3 . F u rthermore, | w 2 ( Q ) − w 2 ( Q ′ ) | ≤ 1 since w 2 ( e ) ≤ 1 for all edges. W e ha ve w 2 ( Q ) + w 2 ( Q ′ ) = 3 − δ 2 . Thus, m in { w 2 ( Q ) , w 2 ( Q ′ ) } ≥ 3 − δ 2 2 − 1 2 ≥ 1 − δ 2 2 . Th is implies w 2 ( P ′ ∪ Q ) ≥ 1 and w 2 ( P ′ ∪ Q ′ ) ≥ 1. Hence, with resp ect to w 2 and w 3 , b oth P ′ ∪ Q and P ′ ∪ Q will do. The fir s t ob jectiv e w 1 remains to b e considered. W e h a ve max { w 1 ( Q ) , w 1 ( Q ′ ) } ≥ 3 − δ 1 2 . Cho osing either P = P ′ ∪ Q or P = P ′ ∪ Q ′ results in w 1 ( P ) ≥ δ 1 + 3 − δ 1 2 ≥ 1. F or undir ec ted graphs and k = 2, w e do not need the assumption th at the w eigh t of eac h edge is at most α u 2 times the wei ght of the cycle cov er. Lemma 3.3 b elo w immediately yields a (1 / 2 − ε ) approximat ion for bi-criteria Max-STSP: First, we compute a P areto curv e of cycle co v ers. Second, w e decomp ose eac h cycle co ver to obtain a collection of paths, whic h we then connect to form Hamiltonian cycles. The follo wing lemma can also b e generalized to arbitrary k (Lemma 3.6). Lemma 3.3. F or every undir e cte d cycle c over C with e dge weights w = ( w 1 , w 2 ) , ther e exists a c ol le ction P ⊆ C of p aths with w ( P ) ≥ w ( C ) / 2 . Pr o of. Let c b e a cycle of C consisting of edges e 1 , e 2 , e 3 . S ince we hav e thr ee edges, there exists one edge e j that is neither the m aximum-w eigh t edge w ith resp ect to w 1 nor the maxim um- w eigh t edge with resp ect to w 2 . W e remo v e this edge. Thus, we ha ve remov ed at most half of c ’s w eight with resp ect to either ob jectiv e. Consequen tly , w e ha v e kept at least half of c ’s w eigh t, whic h pr o ves α u 2 ≥ 1 / 2. 3.1.2 Directed C ycle Co vers F or d irect ed cycle co vers, our aim is again to show that the probabilit y of having not enough w eigh t in one comp onent is less than 1 /k . Ho effding’s inequalit y works only for k ≥ 7. W e use a different approac h, whic h immediately giv es us the desired result for k ≥ 6, and which can b e tw eaked to w ork also for small k . Theorem 3.4. F or al l k ≥ 2 , we have α d k ≥ 1 k +1 . Pr o of. As argued ab o ve, we can restrict our selv es to cycle co vers consisting solely of cycles of length t wo. W e scale the edge weigh ts to ac h iev e w i ( C ) = k + 1 f or all i ∈ [ k ]. This im p lies w i ( e ) ≤ 1 f or all edges e ∈ C . Of every cycle, w e randomly c ho ose one of th e t wo edges and p ut it in to P . Fix an y i ∈ [ k ]. Our aim is to sh o w th at P ( w i ( P ) < 1) < 1 /k , which wo uld p ro v e th e existence of an α d k - decomp osition. Let c 1 , . . . , c m b e the cycles of C with c j = ( e j , f j ). Let w i ( e j ) = a j and w i ( f j ) = b j . W e assume a j ≤ b j for all j ∈ [ m ]. Let δ = P m j =1 a j . T hen, no matter wh ic h edges we c ho ose, w e obtain a we ight of at least δ . Hence, if δ ≥ 1, we are done. O therwise, w e ha ve δ < 1 and replace b j b y b j − a j and a j b y 0. Th en we only need additional w eight 1 − δ , and our new goal is to pro ve P  w i ( P ) < 1 − δ  < 1 /k . This b oils d o wn to th e f ollo win g rand om exp erimen t: W e ha ve n umb ers b 1 , . . . , b m ∈ [0 , 1] with P m j =1 b j = k + 1 − 2 δ . Then w e c ho ose a set I ⊆ [ m ] uniformly at ran d om. F or suc h an I , we d efine (b y abusing notation) w ( I ) = P j ∈ I b j . W e hav e to sho w P  w ( I ) < 1 − δ  < 1 /k . T o this aim, let C 1 , . . . , C z ⊆ [ m ] with z =  k +1 2  b e pairwise disjoin t sets with w ( C ℓ ) ∈ [1 − δ, 2 − δ ). Suc h sets exist: W e select arbitrary elemen ts for C 1 unt il w ( C 1 ) ∈ [1 − δ , 2 − δ ). This can alw a ys b e d one since b j ≤ 1 for all j . Then we con tinue with C 2 , C 3 , and so on. If w e hav e already z − 1 such sets, then w ( C 1 ∪ . . . ∪ C z − 1 ) ≤ (2 − δ ) · ( z − 1) ≤ (2 − δ ) · k 2 ≤ k − δ 6 since k ≥ 2. Thus, at least weig ht k + 1 − 2 δ − ( k − δ ) = 1 − δ is left, wh ic h su ffi ce s for C z . The sets C 1 , . . . , C z do not n ecessarily form a partition of [ m ]. Let C ′ = [ m ] \ ( C 1 ∪ . . . ∪ C z ). W e will ha ve to consider C ′ once in the end of the pro of. No w consider any I , J ⊆ [ m ]. W e say that I ∼ J if I = J △ C ℓ 1 △ C ℓ 2 △ . . . △ C ℓ y for some C ℓ 1 , . . . , C ℓ y . Here, △ denotes the sy m metric difference of sets. The relation ∼ is an equiv alence relation that partitions all subsets of [ m ] into 2 m − z equiv alence classes, eac h of cardinalit y 2 z . Let [ I ] = { J ⊆ [ m ] | J ∼ I } . Lemma 3.5. F or every I ⊆ [ m ] , ther e ar e at most two sets J ∈ [ I ] with w ( J ) < 1 − δ . Pr o of. Without loss of generalit y assu me that w ( I ) = m in J ∈ [ I ] w ( J ). If w ( I ) ≥ 1 − δ , then there is nothing to show. Otherwise, consider any J = I △ C ℓ 1 △ . . . △ C ℓ y ∈ [ I ] with y ≥ 2: w ( J ) ≥ y X p =1 w ( C ℓ p \ I ) ≥ y X p =1 w ( C ℓ p ) | {z } ≥ y · (1 − δ ) ≥ 2 − 2 δ − y X p =1 w ( C ℓ p ∩ I ) | {z } ≤ w ( I ) < 1 − δ ≥ 1 − δ. W e conclude th at the only p ossibilit y f or other s ets J ∈ [ I ] with w ( J ) < 1 − δ is J = I △ C ℓ for some ℓ . W e p ro ve that there is at most one suc h set b y contradict ion. So assume that ther e are J 1 = I △ C 1 and J 2 = I △ C 2 with w ( J 1 ) , w ( J 2 ) < 1 − δ . Then w ( J 1 ) ≥ w ( C 1 ) − w ( C 1 ∩ I ) + w ( C 2 ∩ I ) and w ( J 2 ) ≥ w ( C 2 ) − w ( C 2 ∩ I ) + w ( C 1 ∩ I ). T h us, 2 − 2 δ > w ( J 1 ) + w ( J 2 ) ≥ w ( C 1 ) + w ( C 2 ) ≥ 2 − 2 δ , a con tradiction. A consequence of L emma 3.5 is P  w ( I ) < 1 − δ  < 2 − z +1 = 2 −⌈ k +1 2 ⌉ +1 . T his is less than 1 /k for k ≥ 6. The cases k ∈ { 2 , 3 , 4 , 5 } need sp ecial treatmen t. Let us treat k ∈ { 2 , 4 } first. Here 2 −⌈ k +1 2 ⌉ +1 = 1 /k , wh ic h is almost go o d enough. T o pr o ve P  w ( I ) < 1 − δ  < 1 /k , w e only hav e to fin d a set I suc h th at at most one set J ∈ [ I ] has w ( J ) < 1 − δ . W e claim that ∅ is suc h a set: Of course w ( ∅ ) = 0 < 1 − δ . Bu t for any other J ∈ [ ∅ ], we hav e J = ∅△ C ℓ 1 △ . . . △ C ℓ y = C ℓ 1 ∪ . . . ∪ C ℓ y for some C ℓ 1 , . . . , C ℓ y . The latter equalit y holds since the sets C 1 , . . . , C z are disjoin t. T h us w ( J ) ≥ y · (1 − δ ) ≥ 1 − δ . T o fin ish the pro of, we consid er the case k ∈ { 3 , 5 } . F or this purp ose, we consider I and I = [ m ] \ I sim ultaneously . The classes [ I ] and [ I ] are disjoint : [ I ] = [ I ] would imply C ′ = ∅ . Then P z ℓ =1 w ( C ℓ ) = k + 1 − 2 δ . Since k is o dd, w e ha v e z = k +1 2 . T h us, since k + 1 ≥ 4, there m ust exist an ℓ with w ( C ℓ ) ≥ k + 1 − 2 δ k +1 2 ≥ ( k + 1) · (2 − δ ) k + 1 = 2 − δ, whic h cont radicts w ( C ℓ ) < 2 − δ . 7 W e sh o w that the num b er of sets J ∈ [ I ] ∪ [ I ] with w ( J ) < 1 − δ is at m ost tw o. This w ould p ro ve th e result for k ∈ { 3 , 5 } since this would imp ro ve the b oun d to P ( w ( I ) < 1 − δ ) < 2 − ( z +1)+1 = 2 −⌈ k +1 2 ⌉ < 1 /k . If we had m ore than t wo sets J ∈ [ I ] ∪ [ I ] with w ( J ) < 1 − δ , w e can assume that we ha ve t w o such sets in [ I ]. (W e cannot ha v e more than t w o such J d ue to Lemma 3.5.) W e assume that these t wo sets are I and I ′ = I △ C 1 . No w consid er an y J ∈ [ I ]. Since k is o dd and k + 1 is ev en, we h a ve w ( J ) ≤ z X ℓ =2 w ( C ℓ ) | {z } ≤ (2 − δ ) · ( z − 1) + max { w ( I ) , w ( I ′ ) } | {z } < 1 − δ < (2 − δ ) ·  k + 1 2  − 1 = (2 − δ ) ·  k + 1 2  − 1 = k − k + 1 2 · δ ≤ k − δ. Th u s, all sets J ∈ [ I ] h a ve a w eight of less than k − δ . This implies w ( J ) = k + 1 − 2 δ − w ( J ) > 1 − δ for all J ∈ [ I ]. Thus, if [ I ] conta ins tw o sets whose w eigh t is less th an 1 − δ , then [ I ] con tains n o such set. 3.1.3 Impro v ements and Generalizations T o conclude this section, let us discuss some improv ements of the resu lts of this section. First, as a generalizatio n of Lemma 3.3, cycle co ve rs without cycles of length at most k can b e 1 / 2- decomp osed. This, ho we ver, do es not immediately yield an appro ximation algorithm since finding maxim u m -w eight cycle co vers where eac h cycle m ust h a ve a length of at least k is NP- and APX-hard for k ≥ 3 in directed graphs and for k ≥ 5 [7]. Lemma 3.6. L et k ≥ 1 , and let C b e an arbitr ary cycle c over such that the length of every cycle is at le ast k + 1 . Then ther e exists a c ol le ction P ⊆ C of p aths with w ( P ) ≥ w ( C ) / 2 . Pr o of. T h e pro of is similar to the p roof of Lemma 3.3. Let c b e an y cycle of C . F or eac h i ∈ [ k ], we choose one edge of c that maximizes w i for inclus ion in P . Since c has at least k + 1 edges, this lea v es us (at least) one edge for remo v al. Figure 1 sho ws that T h eorems 3.2 and 3.4, r esp ective ly , are tight for k = 2. Due to these limitations for k = 2, pro ving larger v alues for α u k or α d k do es not immediately yield b etter appro ximation r ati os (see Section 5). H ow ever, for larger v alues of k , Ho effding’s inequalit y yields the existence of Ω(1 / log k )-decomp ositions. T ogether with a differen t tec hn ique for hea vy-w eight cycle co vers, this m igh t lead to impr o ved app ro ximation algorithms for larger v alues of k . Lemma 3.7. We have α u k , α d k ∈ Ω (1 / log k ) . Pr o of. Let A = c ln k + d for some su fficien tly large constan ts c and d . Since α u k ≥ α d k , we can restrict ourselv es to directed graphs. Using the notation of Th eorem 3.2, w e ha ve to sh o w that P ( X < 1) < 1 /k , wher e X = P m j =1 X j and X j assumes v alues in th e interv al [ a j , b j ], b j ≤ a j + 1, P m j =1 ( b j + a j ) 2 ≤ A , and E ( X ) = A/ 2. W e use Ho effding’s in equalit y and plug in t = A/ 2 − 1: P ( X < 1) ≤ exp − 2( A 2 − 1) 2 A ! = exp  − A 2 + 2 − 2 A  < 1 k . 8 (1 , 0) (0 , 1) (1 , 0) (0 , 1) (1 , 0) (0 , 1) (a) α d 2 ≤ 1 / 3. (0 , 1) (1 , 0) (1 , 1) (b) α u 2 ≤ 1 / 2. Figure 1: Examples that limit the p ossibilit y of decomp ositio n. P ← Decomp ose ( C , w , k , α ) input: cycle co v er C , edge we ights w , k ≥ 2, w ( e ) ≤ α · w ( C ) for all e ∈ C output: a collecti on P of paths 1: obtain w ′ from w by scaling eac h comp onen t such that w ′ i ( C ) = 1 /α for all i 2: n orm ali ze C to C ′ as describ ed in the text suc h that C ′ consists solely of cycles of length three (undir ec ted) or t wo (directed) 3: w hile there are cycles c and c ′ in C ′ with w ′ ( c ) ≤ 1 / 2 and w ′ ( c ′ ) ≤ 1 / 2 do 4: com bine c and c ′ to ˜ c w ith w ′ (˜ c ) = w ′ ( c ) + w ′ ( c ′ ) 5: replace c and c ′ b y ˜ c in C ′ 6: try all p ossible com binations of d eco mp ositions 7: choose one P ′ that maximizes min i ∈ [ k ] w ′ i ( P ) 8: translate P ′ ⊆ C ′ bac k to obtain a d ecomp osition P ⊆ C 9: retur n P Algorithm 1: A deterministic algorithm for find ing a decomp osition. 3.2 Finding Decomp ositions While w e kno w that d ec omp ositions exist due to the previous section, w e ha ve to find them efficien tly in ord er to use them in our appro ximation algorithm. W e present a deterministic algorithm and a faster randomized algorithm for find ing d eco mp ositions. 3.2.1 Deterministic Algorithm Decompose (Algorithm 1) is a deterministic algorithm for find ing a decomp osition. The idea b ehind this algorithm is as f oll ows: First, we scale the w eigh ts suc h that w ( C ) = 1 /α . Then w ( e ) ≤ 1 f or all edges e ∈ C . Second, we normalize all cycle co vers s uc h that they consist solely of cycles of length t wo (in case of directed graph s ) or th ree (in case of undir ect ed graphs). Third, we com bin e very light cycles as long as p ossible. More pr eci sely , if there are t w o cycles c and c ′ suc h that w ′ ( c ) ≤ 1 / 2 and w ′ ( c ′ ) ≤ 1 / 2, we com b ine them to one cycle ˜ c with w ′ (˜ c ) ≤ 1. T he requiremen ts for an α -d ec omp osition to exist are still fu lfilled. F urthermore, an y α -decomp osition of C ′ immediately yields an α -decomp osition of C . The p roof of the follo w ing lemma follo w s immediately from the existence of d ec omp ositions (Theorems 3.2 and 3.4). Lemma 3.8. L et k ≥ 2 . L et C b e an undir e cte d cycle c over and w 1 , . . . , w k b e e dge weights such that w ( e ) ≤ α u k · w ( C ) . Then Dec ompose ( C, w , k , α u k ) r eturns a c ol le ction P of p aths with w ( P ) ≥ α u k · w ( C ) . 9 P ← Ran dDecompose ( C , w , k , α ) input: cycle co v er C , edge we ights w = ( w 1 , . . . , w k ), k ≥ 2, w ( e ) ≤ α · w ( C ) for all e ∈ C output: a collecti on P of paths with w ( P ) ≥ α · w ( C ) 1: if k ≥ 6 then 2: rep eat 3: randomly c ho ose one edge of ev ery cycle of C 4: remo v e the c hosen edges to obtain P 5: un til w ( P ) ≥ α · w ( C ) 6: else 7: P ← Decomp ose ( C, w , k , α ) Algorithm 2: A randomized algorithm for findin g a decomp osition. L et C b e a dir e cte d cycle c over and w 1 , . . . , w k b e e dge weights such that w ( e ) ≤ α d k · w ( C ) . Then Decompose ( C, w, k , α d k ) r eturns a c ol le ction P of p aths with w ( P ) ≥ α d k · w ( C ) . Let us also estimate the runn ing-t ime of Deco mpose . Th e n orm ali zation in lines 1 to 5 can b e implemented to ru n in linear time. Du e to the normalization, the weig ht of ev ery cycle is at least 1 / 2 with resp ect to at least one w ′ i . Thus, we ha v e at most 2 k/α u k cycles in C ′ in the undirected case and at most 2 k /α d k cycles in C ′ in the directed case. In either case, w e ha v e O ( k 2 ) cycles. All of these cycles are of length t wo or of length three. Thus, w e find an optimal decomp osition, wh ic h in particular is an α u k or α d k -decomp ositio n , in time linear in the input size and exp onen tial in k . 3.2.2 Randomized Algorithm By exp lo iting the probabilistic argum en t of the pr evious section, we can find a d ecomp osition m uch faster w ith a randomized algorithm. Ra ndDecompose (Algorithm 2) do es th is: W e c ho ose the edges to b e deleted u niformly at r an d om for every cycle. T h e p r obabilit y that w e obtain a decomp osition as required is p ositiv e and b ounded from b elo w b y a constan t. F urthermore, as the p roofs of Theorems 3.2 and 3.4 show, this p r obabilit y tends to one as k increases. F or k ≥ 6, it is at least approximat ely 0 . 7 for undir ec ted cycle co vers and at least 1 / 4 for directed cycle co ve rs. F or k < 6, we ju st use our deterministic algorithm, whic h h as linear runn in g-t ime for constan t k . The follo wing lemma follo ws f rom the considerations ab o v e. Lemma 3.9. L et k ≥ 2 . L et C b e an undir e cte d cycle c over and w 1 , . . . , w k b e e dge weights such that w ( e ) ≤ α u k · w ( C ) . Then R andDecompose ( C, w , k , α u k ) r eturns a c ol le ction P of p aths with w ( P ) ≥ α u k · w ( C ) . L et C b e a dir e cte d cycle c over and w 1 , . . . , w k b e e dge weights such that w ( e ) ≤ α d k · w ( C ) . Then RandDecompos e ( C, w, k , α d k ) r eturns a c ol le ction P of p aths with w ( P ) ≥ α d k · w ( C ) . The exp e c te d running-time of RandDeco mpose is O ( | C | ) . 4 Appro ximation Algorithms Based on the idea sk etc hed in Section 2, w e can no w present our approximat ion algorithms for m u lti-criteria Max-A TSP and Max-STSP . Ho w eve r, in p articular for Max-STSP , some additional w ork h as to b e done if h ea vy ed ges are presen t. 10 P TSP ← MC-MaxA TSP ( G, w , k , ε ) input: directed complete graph G = ( V , E ), k ≥ 1, edge we ights w : E → N k , ε > 0 output: appro ximate P areto curve P TSP for k -criteria Max-TSP 1: if k = 1 then 2: compute a 2 / 3 appro ximation P TSP 3: else 4: compute a (1 − ε ) appro ximate Pareto curv e C of cycle co v ers 5: P TSP ← ∅ 6: for all cycle cov ers C ∈ C do 7: if w ( e ) ≤ α d k · w ( C ) for all ed ge s e ∈ C t hen 8: P ← Decomp ose ( C , w , k ) 9: add edges to P to form a Hamiltonian cycle H 10: add H to P TSP 11: else 12: let e = ( u, v ) ∈ C b e an edge with w ( e ) 6≤ α d k · w ( C ) 13: for all a, b, c, d ∈ V suc h that P e a,b,c,d is legal do 14: for i ← 1 to k do 15: obtain G ′ from G b y cont racting the paths of P e a,b,c,d 16: obtain w ′ from w b y removing the i th ob jectiv e 17: P ′ TSP ← MC-MaxA TSP ( G ′ , w ′ , k − 1 , ε ) 18: for all H ′ ∈ P ′ TSP do 19: form a Hamiltonian cycle from H ′ plus P e a,b,c,d ; ad d it to P TSP 20: form a Hamiltonian cycle from H ′ plus ( u, v ); add it to P TSP Algorithm 3: Appro ximation algorithm for k -criteria Max-A TSP . 4.1 Multi-Criteria Max-A TSP W e fi rst pr esen t our algorithm for Max-A TS P (Algorithm 3) since it is a bit easier to analyze. First of all, we compute a (1 − ε ) app ro ximate Pa reto cur v e C of cycle co ve rs . Then, for ev ery cycle co ver C ∈ C , w e decide whether it is a ligh t-we ight cycle cov er or a hea vy-w eight cycle co v er (line 7). If C h as only light-w eight ed ge s, we decomp ose it to obtain a collec tion P of paths. Then we add edges to P to obtain a Hamiltonian cycle H , which we then add to P TSP . If C con tains a h ea vy-weig ht edge, then there exists an edge e = ( u, v ) and an i with w i ( e ) > α k · w i ( C ). W e pic k on e suc h ed ge. Then we iterate o ve r all p ossible vertices a, b, c, d (including equalities and including u and v ). W e denote by P e a,b,c,d the graph with ve r tices u , v , a , b , c , d and edges ( a, u ), ( u, b ), ( c, v ), and ( v , d ). W e call P e a,b,c,d le gal if it can b e extended to a Hamiltonian cycle: P e a,b,c,d is legal if and only if it consists of one or t wo verte x-disj oin t directed paths. Figure 2 shows the different p ossibilities. F or ev ery legal P e a,b,c,d , w e contrac t the paths as follo ws: W e remov e all outgoing edges of a and c , all incoming edges of b and d , and all edges incident to u or v . T hen we id entify a and b as we ll as c and d . If P e a,b,c,d consists of a single path, then we remo ve all ve rtices except the t w o endp oin ts of this path, and w e identify these t wo end p oin ts. In this w a y , we obtain a sligh tly s m all er instance G ′ . Then, for every i , w e remo ve th e i th ob jectiv e to obtain w ′ , and recurs e on G ′ with only k − 1 ob jectiv es w ′ . T his y ields app r o ximate P areto curves P ′ TSP of Hamiltonian cycles of G ′ . No w consid er an y H ′ ∈ P ′ TSP . W e exp an d 11 b a u v d c (a) Tw o disjoint paths. a u b = c v d (b) O ne pa th with an intermediate vertex. a u = c v = b d (c) O ne path including e . Figure 2: The three p ossib ilities of P e a,b,c,d . Symmetrically to (b ), w e also ha v e a = d . Sym- metrically to (c), we also ha ve v = a and u = d . the con tracted paths to obtain H . Then we constru ct tw o tours: First, we ju st add P e a,b,c,d to H , whic h yields a Hamiltonian cycle b y construction. S ec ond , we observ e that no edge in H is incident to u or v . W e add the edge ( u, v ) to H as well as some more edges such that we obtain a Hamiltonian cycle. W e p ut the Hamiltonian cycles thus constructed into P TSP . W e h a ve not ye t discussed the success pr ob ab ility . Randomness is needed for compu ting the appr o ximate Pa reto cur ves of cycle co v ers and the recursiv e calls of MC-MaxA TSP with k − 1 ob jectiv es. Let N b e the size of the instance at hand , and let p k ( N , 1 /ε ) is a p olynomial that b oun ds the size of a (1 − ε ) approximat e Paret o curv e from ab o v e. T hen we n eed at most N 4 · p k ( N , 1 /ε ) r ec u rsiv e calls of M C- Max A TSP . In total, the n u m b er of calls of randomized algorithms is b oun ded by some p olynomial q k ( N , 1 /ε ). W e amplify the success probabilities of these calls such that the p robabilit y is at least 1 − 1 2 · q k ( N , 1 /ε ) . Thus, the probab ility that one suc h call is not su cce ssfu l is at most q k ( N , 1 /ε ) · 1 2 · q k ( N , 1 /ε ) ≤ 1 / 2 b y a u nion b oun d. Hence, the success probabilit y of the algorithm is at least 1 / 2. Instead of Decompo se , w e can also u se its randomized count erpart Rand Decompose . W e mo dify RandDec ompose such that the run ning-time is guaran teed to b e p olynomial and that there is only a small p robabilit y that Ra ndDecompose err s. F u r thermore, we hav e to mak e the error p robabilities of the cycle co v er computation as wel l as the recursiv e calls of MC-MaxA TSP sligh tly smaller to main tain an o verall success probabilit y of at least 1 / 2. Ov erall, the run ning-time of MC-MaxA TSP is p olynomial in the inp ut size and 1 /ε , whic h can b e s een b y indu ct ion on k : W e hav e a p olynomial time appr o ximation algorithm for k = 1. F or k > 1, the appro ximate P areto cu r v e of cycle co v ers can b e compu ted in p olynomial time, yielding a p olynomial num b er of cycle cov ers. All fur ther computations can also b e implemen ted to r un in p olynomial time since MC-MaxA TSP for k − 1 r uns in p olynomial time b y indu ctio n hyp othesis. Theorem 4.1. MC-MaxA TSP is a r andomize d 1 k +1 − ε appr oximation for multi-criteria Max-A TSP. Its running- time is p olynomial in the input size and 1 /ε . Pr o of. W e ha ve already discuss ed the err or p robabilitie s and the r unning-time. Thus, it re- mains to consider the appro ximation ratio, and we can assume in the follo wing, that all ran- domized computations are successful. W e pr o ve the theorem by in duction on k . F or k = 1, this follo ws sin ce mono-criterion Max-A TSP can b e appr o ximated w ith a factor 2 / 3 > 1 / 2. No w assu me th at the theorem holds for k − 1. W e hav e to pr ov e that, for ev ery Hamiltonian cycle ˆ H , there exists a Hamiltonian cycle H ∈ P TSP with w ( H ) ≥  1 k +1 − ε  · w ( ˆ H ). Since eve r y Hamiltonian cycle is in particular a cycle cov er, there exists a C ∈ C with w ( C ) ≥ (1 − ε ) · w ( ˆ H ). No w we distinguish t w o cases. The first case is that C consists solely of ligh t-w eight edges, i. e., w ( e ) ≤ 1 k +1 · w ( C ), then 12 Decompose returns a collecti on P of p at h s with w ( P ) ≥ 1 k +1 · w ( C ) ≥  1 k +1 − ε  · w ( ˆ H ), whic h yields a Hamiltonian cycle H with w ( H ) ≥ w ( P ) ≥  1 k +1 − ε  · w ( ˆ H ) as claimed. The second case is that C con tains at least one h ea vy-weig ht edge e = ( u, v ). Let ( a, u ), ( u, b ), ( c, v ), and ( v , d ) b e th e edges in ˆ H th at are incident to u or v . (W e m a y h av e some equalities among the v ertices as shown in Figure 2.) Note that ˆ H do es not necessarily con tain the edge e . W e consider the corresp ond ing P e a,b,c,d and divide the second case in to tw o sub cases. The first su b ca se is that there exists a j ∈ [ k ] with w j ( P e a,b,c,d ) ≥ 1 k +1 · w j ( ˆ H ), i. e., at least a 1 k +1 fraction of the j th ob jectiv e is concen trated in P e a,b,c,d . (W e can hav e j = i , but this is not necessarily the case.) Let J ⊆ [ k ] b e the set of suc h j . W e fix one j ∈ J arb itrarily and consider the graph G ′ obtained by remo ving the j th ob jectiv e and con tracting the paths ( a, u, b ) and ( c, v , d ). A fr ac tion of 1 − 1 k +1 = k k +1 of the w eigh t of ˆ H is left in G ′ with resp ect to all ob j ective s but those in J . T hus, G ′ con tains a Hamiltonia n cycle ˆ H ′ with w ℓ ( ˆ H ′ ) ≥ k k +1 · w ℓ ( ˆ H ) for all ℓ ∈ [ k ] \ J . Sin ce ( k − 1)-c riteria Max-A TS P can b e approximat ed with a factor of 1 k − ε by assu mption, P ′ TSP con tains a Hamiltonian cycle H ′ with w ℓ ( H ′ ) ≥ ( 1 k − ε ) · k k +1 · w ℓ ( ˆ H ) ≥  1 k +1 − ε  · w ℓ ( ˆ H ) for all ℓ ∈ [ k ] \ J . T ogether with P e a,b,c,d , whic h con tribu tes enou gh weig ht to the ob jectiv es in J , we obtain a Hamiltonian cycle H with w ( H ) ≥  1 k +1 − ε  · w ( ˆ H ), whic h is as claimed. The second sub case is that w j ( P e a,b,c,d ) ≤ 1 k +1 · w j ( H ) for all j ∈ [ k ]. Thus, at least a fraction of k k +1 of the weig ht of ˆ H is outside of P e a,b,c,d . W e consider the case with the i th ob jectiv e remo v ed. Th en, with the same argument as in the fi rst sub case, w e obtain a Hamiltonian cycle H ′ of G ′ with w ℓ ( H ′ ) ≥  1 k +1 − ε  · w ℓ ( ˆ H ) for all ℓ ∈ [ k ] \ { i } . T o obtain a Hamiltonian cycle of G , we take th e edge e = ( u, v ) and connect its end p oin ts appropriately . (F or instance, if a, b, c, d are distinct, then we add th e path ( a, u, v , d ) and th e edge ( c, b ).) This yields enough wei ght for the i th ob jectiv e in order to obtain a Hamiltonian cycle H with w ( H ) ≥  1 k +1 − ε  · w ( ˆ H ) since w i ( e ) ≥ 1 k +1 · w ( C ) ≥  1 k +1 − ε  · w ( ˆ H ). 4.2 Multi-Criteria Max-STSP MC-MaxA TSP wo rk s of course also f or u ndirected graphs, for which it ac hieve s an appro xi- mation ratio of 1 k +1 − ε . But w e can d o b etter for undirected graphs. Our algorithm MC-MaxSTSP for undir ec ted graph s (Algorithm 4) starts by computin g an appro ximate Pa reto cur v e of cycle co vers just as MC-MaxA TSP d id. T hen we consid er eac h cycle co ver C separately . I f C consists solely of ligh t-w eigh t ed ges, then we can decomp ose C using Decomp ose . If C con tains one or more hea vy-w eigh t ed ge s, then some m ore w ork has to b e done than in the case of d irect ed graphs. The reason is that we cannot s imply con tract paths – this w ould make the n ew graph G ′ (and the edge w eight s w ′ ) asymmetric. So assume that a cycle co v er C ∈ C con tains a heavy- weigh t edge e = { u, v } . L et i ∈ [ k ] b e suc h th at w i ( e ) ≥ w i ( C ) /k . In a firs t attempt, w e remo ve th e i th ob jectiv e to obtain w ′ . Then w e set w ′ ( f ) = 0 for all edges f incident to u or v . W e recurse with k − 1 ob jectiv es on G w ith edge weigh ts w ′ . T his yields a tour H ′ on G . No w we remo ve all edges incident to u or v of H ′ and add new edges includin g e . I n this w a y , w e get enough w eigh t w ith resp ect to ob jectiv e i . Unfortunately , there is a p r oblem if th er e is an ob jectiv e j and an edge f incident to u or v su c h that f con tains almost all weig ht with resp ect to w j : W e cannot guaran tee that this edge f is included in H without further mo difying H ′ . T o cop e with this problem, we d o the follo wing: In add itio n to u and v , we set the w eigh t of all edges incident to the other ve rtex of f to 0. 13 P TSP ← MC-MaxSTSP ( G, w , k , ε ) input: und irecte d complete graph G = ( V , E ), k ≥ 2, edge wei ghts w : E → N k , ε > 0 output: appro ximate P areto curve P TSP for k -criteria Max-TSP 1: compu te a (1 − ε ) appr o ximate P areto curv e C of cycle co v ers 2: P TSP ← ∅ 3: if k = 2 then 4: for all C ∈ C do 5: P ← Decompose ( C, w , k ) 6: add edges to P to form a Hamiltonian cycle H 7: add H to P TSP 8: else 9: for all cycle cov ers C ∈ C do 10: if w ( e ) ≤ w ( C ) /k for edges e ∈ C then 11: P ← Decompo se ( C, w , k ) 12: add edges to P to form a Hamiltonian cycle H 13: add H to P TSP 14: else 15: let i ∈ [ k ] and e = { u, v } ∈ C with w i ( e ) > w i ( C ) /k 16: for all ℓ ∈ { 0 , . . . , 4 k } , distinct x 1 , . . . , x ℓ ∈ V \ { u, v } , and k ∈ [ k ] do 17: U ← { x 1 , . . . , x ℓ , u, v } 18: obtain w ′ from w b y removing the j th ob jectiv e 19: set w ′ ( f ) = 0 for all edges f incident to U 20: P U,j TSP ← MC-MaxS TSP ( G, w ′ , k − 1 , ε ) 21: for all H ∈ P U,j TSP do 22: remo v e all edges f from H with f ⊆ U to obtain H ′ 23: for all H U suc h that H ′ ∪ H U is a Hamiltonian cycle do 24: add H ′ ∪ H U to P TSP Algorithm 4: Appro ximation algorithm for k -criteria Max-STS P . Then we r ecur se. Unfortunately , there may b e another ob jectiv e j ′ that no w causes p r oblems. T o solv e the whole problem, w e iterate o v er all ℓ = 0 , . . . , 4 k and ov er all additional vertices x 1 , . . . , x ℓ 6 = u, v . Let U = { x 1 , . . . , x ℓ , u, v } . W e r emov e one ob jectiv e i ∈ [ k ] to obtain w ′ , s et the weig ht of all edges in ciden t to U to 0, and recurse with k − 1 ob j ec tive s. Although the time needed to do this is exp onen tial in k , w e mainta in p olynomial r unning-time for fix ed k . As in the case of d irect ed graph s , w e can make the success pr ob ab ility of ev ery randomized computation small enough to mainta in a success probabilit y of at least 1 / 2. The base case is no w k = 2: In this case, eve ry cycle co v er p ossesses a 1 / 2 decomp osition, and we do not ha v e to care ab out hea vy-w eigh t ed ges. Overall , we obtain the follo wing result. Theorem 4.2. MC-MaxSTSP is a r andomize d 1 k − ε appr oximation for multi-criteria M ax- STSP. Its running- time is p olynomial in the input size and 1 /ε . Pr o of. W e ha ve already d ea lt with er r or pr ob ab ilities and run ning-time. Thus, w e can assum e that all randomized computations are successful in the follo win g. What remains to b e analyzed is the appro ximation ratio. As in the pro of of Th eo rem 4.1 , the pr oof is b y ind u ctio n on k . The base case is k = 2. Let ˆ H b e an arbitrary Hamiltonian cycle. Then there is a C ∈ C with w ( C ) ≥ (1 − ε ) · w ( ˆ H ). F r om C , w e obtain a Hamiltonian cycle H with w ( H ) ≥ 1 2 · w ( C ) ≥ 14 ( 1 2 − ε ) · w ( ˆ H ) by decomp osition and Lemma 3.3. Let us analyze M C- Max STSP for k ≥ 3 ob jectiv es. By the indu cti on hyp othesis, w e can assume that MC-MaxSTSP is a 1 k − 1 − ε appro ximation for ( k − 1)-criteria Max-STSP . Let ˆ H b e any Hamiltonian cycle. W e hav e to show that P TSP con tains a Hamiltonian cycle H w ith w ( H ) ≥  1 k − ε  · w ( ˆ H ). There is a C ∈ P TSP with w ( C ) ≥ (1 − ε ) · w ( ˆ H ). W e ha v e to d istinguish t wo cases. First, if C consists solely of ligh t-we ight edges, i. e., w ( e ) ≤ w ( C ) /k for all e , then we obtain a Hamiltonian cycle H from C with w ( H ) ≥ w ( C ) /k ≥  1 k − ε  · w ( ˆ H ). Second, let e ∈ C and i ∈ [ k ] with w i ( e ) > w i ( C ) /k . W e construct sets I ⊆ [ k ], X ⊆ ˆ H , and U ⊆ V in ph ase s as follo w s (w e do not actually construct th ese sets, but only need them for the analysis): I nitial ly , I = X = ∅ and U = { u, v } . In ev ery phase, w e consider the s et X ′ of all edges of ˆ H that ha ve exactly on e en d p oin t in U . W e alw ays ha ve | X ′ | ≤ 4 by construction. Let I ′ = { j ∈ [ k ] | j / ∈ I , w j ( X ′ ) ≥ w j ( ˆ H ) /k } . If I ′ is emp t y , then we are done. Otherw ise, add I ′ to I , add X ′ to X , and add all n ew en dp oin ts of vertic es in X ′ to U . W e add at least one elemen t to I in ev ery phase. Thus, | X | ≤ 4 k and | U | ≤ 4 k + 2 s in ce | I | ≤ k . Let w in = w ( X ), and let w ∂ = P f ∈ ˆ H : | f ∩ U | =1 w ( f ) b e the weigh t of edges of ˆ H that ha ve exactly one endp oint in U . L et w out = w ( ˆ H ) − w in − w ∂ . By construction, we h av e w ∂ j < 1 /k for all j / ∈ I . Oth er w ise, w e would h a ve added more edges to X . W e distinguish t wo sub cases. T he firs t su bcase is that I = ∅ . Then w in = 0 and w ∂ < 1 /k . Consider the set P ∅ ,i TSP and the edge w eigh ts w ′ used to ob tain it. W e h a ve w ′ j ( ˆ H ) = w out j >  k − 1 k  · w j ( ˆ H ) for j 6 = i . By the in duction hyp ot h esis, there is an H ∈ P ∅ ,i TSP with w ′ j ( H ) ≥  1 k − 1 − ε  ·  k − 1 k  · w ( ˆ H ) ≥  1 k − ε  · w ( ˆ H ) for j 6 = i . W e remo ve all ed ge s inciden t to u or v to obtain H ′ . S ince the weigh t of all these edges h as b een set to 0, we hav e w ′ ( H ′ ) = w ′ ( H ). T h ere exists a s et H ∅ suc h that e ∈ H ∅ and H ′ ∪ H ∅ is a Hamiltonian cycle. F or this cycle, w h ic h is in P TSP , w e h a ve w i ( H ′ ∪ H ∅ ) ≥ w i ( e ) ≥ w i ( C ) /k ≥  1 k − ε  · w ( ˆ H ) and, for j 6 = i , w j ( H ′ ∪ H ∅ ) ≥ w ′ j ( H ) ≥  1 k − ε  · w ( ˆ H ) . The second sub case is that I is not empty . Let j ∈ I , and let U . W e consider P U,j TSP . Let w in , w ∂ , and w out b e as defined ab o ve . By the in duction hyp othesis, the set P U,j TSP con tains a Hamiltonian cycle co v er H with w ′ ℓ ( H ) ≥  1 k − 1 − ε  · w out ℓ for ℓ 6 = j . W e remov e all edges inciden t to U from H to obtain H ′ with w ′ ( H ′ ) = w ′ ( H ). By construction H ′ ∪ X is a collectio n of paths. W e add edges to X to obtain H U suc h that H ′ ∪ H U is a Hamiltonian cycle. Let us estimate the w eigh t of H ′ ∪ H U . F or all ℓ ∈ I , we hav e w ℓ ( H ′ ∪ H U ) ≥ w ℓ ( H U ) ≥ w ℓ ( ˆ H ) /k . F or all ℓ / ∈ I , we hav e w ℓ ( H ′ ∪ H U ) ≥ w ′ ℓ ( H ′ ) + w in ℓ ≥  1 k − 1 − ε  · ( w out ℓ + w in ℓ ) ≥  1 k − 1 − ε  · ( w ℓ ( ˆ H ) − w ∂ ℓ ) ≥  1 k − 1 − ε  · k − 1 k · w ℓ ( ˆ H ) ≥  1 k − ε  · w ℓ ( ˆ H ) , whic h completes th e pro of. 15 5 Remarks The analysis of the appro ximation ratios of our algorithms is essenti ally optimal: Ou r approac h can at b est lead to approximati on ratios of 1 k + c for some c ∈ Z . The reason is as follo ws: Assume that ( k − 1)-criteria Max-TSP can b e approximat ed with a factor of τ k . If w e hav e a k -criteria in s ta n ce, we ha ve to set th e threshold for hea vy-w eigh t edges somewh ere. Assume for the moment that this thresh old α k b e arb itrary . Then the r atio for k -criteria Max-TSP is min { α k , (1 − α k ) · τ k − 1 } . Cho osing α k = τ k − 1 τ k − 1 +1 maximizes this ratio. Thus, if τ k − 1 = 1 /T for some T , then τ k ≤ τ k − 1 τ k − 1 +1 = 1 T +1 . W e conclude that the denominator of the appr o ximation ratio increases by at least 1 if we go from k − 1 to k . F or und irect ed graphs, we ha ve obtained a ratio of roughly 1 /k , which is optimal since α u 2 = 1 / 2 imp lie s c ≥ 0. Similarly , for directed graphs, we ha ve a ratio of 1 k +1 , whic h is also optimal since α d 2 = 1 / 3 im p lies c ≥ 1. Due to the existence of Ω(1 / log k )-decomp ositio ns, we conjecture that b oth k -criteria Max- STSP and k -criteria Max-A TSP can in fact b e appro ximated with factors of Ω(1 / log k ). This, ho we ver, requires a different appr oac h or at least a new tec hn iqu e for h eavy-w eight edges. References [1] Ra vindr a K. Ah u ja, Th omas L. Magnan ti, and James B. Orlin. Network Flows: The ory, Algor ithms, and Applic ations . Prentic e-Hall, 1993. [2] Zhi-Zhong C hen, Y uusu k e Ok amoto, and Lu sheng W ang. Imp r o ved deterministic appr o x- imation algorithms for Max TSP. Information Pr o c essing L etters , 95(2):333– 342, 2005. [3] Matthias Ehrgott. Mu lticriteria Optimization . S pringer, 2005. [4] W assily Ho effding. Prob ab ility inequalities for sums of b ounded ran d om v ariables. Journal of the Americ an Statistic al A sso ciation , 58(3 01):13–30, 1963. [5] Haim Kaplan, Moshe Lewe nstein, Nira Sh afrir, and Maxim I. Sviridenko. Appro ximation algorithms for asymmetric T SP by d ec omp osing directed regular multigraphs. Journal of the ACM , 52(4):602– 626, 2005. [6] Bo do Man they . App r o ximate pareto cur v es for the asymmetric tra veling salesman prob- lem. Computing Researc h R ep ository cs.DS/0711.215 7, arXiv, 2007. [7] Bo do Man they . On approxi mating r estrict ed cycle co ve rs . SIAM Journal on Computing , 38(1): 181–206, 2008. [8] Bo do Man they and L. Sh an k ar Ram. Ap pro ximation algorithms for m ulti-criteria tra v eling salesman problems. Algorithmic a , to app ear. [9] Ketan Mulm uley , Umesh V. V azirani, and Vija y V. V azirani. Matc hing is as easy as matrix in version. Combinatoric a , 7(1):105 –113, 1987. [10] Chr isto s H. P apadimitriou and Mihalis Y ann ak akis. On th e approxima b ility of tr ad e-offs and optimal access of web sour ces. In P r o c. of the 41st Ann. IEEE Symp. on F oundations of Computer Scienc e (FOCS) , pages 86–92. IEEE Computer So ciet y , 2000 . 16