Welfare Maximization and the Supermodular Degree

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1 Welfare Maxmzaton and the Supermodular Degree Urel Fege Wezmann Insttute of Scence Ran Izsak Wezmann Insttute of Scence May 29, 2013 Abstract Gven a set of tems and a collecton of players, each wth a nonnegatve monotone valuaton set functon over the tems, the welfare maxmzaton problem requres that every tem be allocated to exactly one player, and one wshes to maxmze the sum of values obtaned by the players, as computed by applyng the respectve valuaton functon to the bundle of tems allocated to the player. Ths problem n ts full generalty s NP-hard, and moreover, at least as hard to approxmate as set packng. Better approxmaton guarantees are known for restrcted classes of valuaton functons. In ths work we ntroduce a new parameter, the supermodular degree of a valuaton functon, whch s a measure for the extent to whch the functon exhbts supermodular behavor. We desgn an approxmaton algorthm for the welfare maxmzaton problem whose approxmaton guarantee s lnear n the supermodular degree of the underlyng valuaton functons. 1 Introducton The welfare maxmzaton problem (also known as combnatoral aucton s the followng. There s a set of players and a set of ndvsble tems. Each player has ts own (monotone non-decreasng valuaton for any subset of tems. The goal s to dstrbute the tems to the players whle maxmzng socal welfare - the sum of values of all players, by ther personal valuatons. The welfare maxmzaton problem s N P-hard to approxmate wth any reasonable guarantee. For ths reason past research consdered restrctons on the class of set functons that may serve as valuaton functons for the players. Lehmann, Lehmann and Nsan [15] consdered complement free functons, whch essentally means that a value of a set of tems cannot exceed the sum of values of ts parts. They presented a herarchy of classes of complement free functons, and establshed constant factor approxmatons for the welfare maxmzaton problem n some cases. 1 Subsequent work establshed constant factor approxmaton for all classes of complement free functons. Ths made t clear that the poor approxmaton guarantees for the general case must come from complementartes ACM, Ths s the authors verson of the work. It s posted here by permsson of ACM for your personal use. Not for redstrbuton. The defntve verson was publshed n proceedngs of the 4th conference on Innovatons n Theoretcal Computer Scence, 1 There was earler work wth related results that used dfferent termnology. For example, Fsher, Nemhauser and Wolsey [10] studed the m-box problem whch s essentally the welfare maxmzaton problem wth monotone submodular valuaton functons. Among ther results there was a greedy approxmaton algorthm for a generalzed verson of ths problem wth approxmaton guarantee 1/2. 1

2 (sets whose value s larger than that of the sum of ther parts. Recently, Abraham, Babaoff, Dughm and Roughgarden [1] consdered a restrcted class of set functons strctly based on complementartes (n partcular, no set s valued less than the sum of ts parts. Among other results, they presented an algorthm wth an approxmaton guarantee that s lnear n a certan parameter related to the extent of these complementartes. In the current work, smlarly to [1], we express the approxmaton guarantees as a functon of some parameter assocated wth the underlyng set functons. The smaller ths parameter s, the better the approxmaton guarantee. However, we depart from the practce of consderng restrcted classes of set functons our parametrzaton can be appled to any set functon. It s most advantageous (n the sense that our approxmaton guarantees are not bad whereas prevous work does not apply to these set functons when the set functons are bascally submodular (offer decreasng margnal value, whch s the dscrete analog of convexty, but exhbt a lmted amount of complementartes. As a smple example of how such functons may arse, consder shoppng for shoes. Each addtonal par of shoes may have decreasng margnal value, but wthn a par of shoes, the left and rght shoe are together worth more than the sum of values of each shoe on ts own. Specfcally, we ntroduce two new (as far as we know complexty measures of set functons. One s the dependency degree. Roughly speakng, ths s the maxmum number of tems that may nfluence the margnal value of any tem wth respect to any possble subset of other tems. The other s the supermodular degree. Roughly speakng, ths s the dependency degree, takng nto account only tems that may ncrease the margnal value of an tem. That s, negatve dependences do not ncrease the latter complexty measure. In partcular, submodular functons mght have arbtrary dependency degree, but ther supermodular degree s 0. Ths measure can also be seen, n a sense, as the degree of complementarty. We desgn two greedy approxmaton algorthms for the welfare maxmzaton problem, each wth approxmaton guarantee lnear n the maxmum of one of these measures over the set functons of the players. 2 Prelmnares We frstly defne formally the welfare maxmzaton problem: Defnton 2.1 An nstance I(P, M, v of the welfare maxmzaton problem s the followng: P s a set of n players 1,...,n. M s a set of m tems j 1,...,j m. v s a vector of n valuaton functons (set functons v 1,...,v n, where v p : 2 M R + s the valuaton functon assocated wth the player p P. 2 For any p P, v p s restrcted to be monotone non-decreasng and wth value 0 for the empty set (and hence also non-negatve. A feasble soluton to I s a mappng SOL : M P, allocatng each of the tems to exactly one player. Ths mappng nduces for each player p P a set SOL p of the tems mapped to her. The utlty/value of a player p P s defned as v p (SOL p. Our am s to maxmze the socal welfare v(sol def = v p (SOL p. p P 2 We use R + to ndcate the set of all non-negatve real numbers (that s, 0 s ncluded. 2

3 2.1 Types of set functons wthout complementartes We recall prevously studed types of set functons wthout complementartes (see for example [15, 8]. Let S be a set and let f : 2 S R + be a set functon. Defnton 2.2 We say that f s submodular f for any S S S and x S \S, f(x S f(x S. Defnton 2.3 We say that f s fractonally subaddtve f for every subset S S, subsets T S and every coeffcents 0 < α 1 such that for any x S, :x T α 1, t holds that f(s α f(t. Defnton 2.4 We say that f s subaddtve or complementfree f for every S 1,S 2 S, f(s 1 S 2 f(s 1 +f(s Measurng dependences In ths secton we ntroduce complexty measures capturng dependences of tems n a groundset of a set functon. For convenence, we treat the set functons as valuaton functons of players of an nstance of the welfare maxmzaton problem (for notatons only. Let M be a set, let v p : 2 M R + be a valuaton functon of player p P and let j M. We frstly recall the followng defnton (see for example [15]: Defnton 2.5 Let p P and let j M. The margnal valuaton functon v p,j : 2 M\{j} R + s a functon mappng each subset S M \{j} to the margnal value of j gven S: v p,j (S def = v p (S {j} v p (S. We denote the margnal value v p,j (S also by v p (j S. For S = {j 1,...,j S } M and S M\S we also use ether of the notatons v p (j 1,...,j S S or v p (S S to ndcate v p (S S v p (S. Defnton 2.6 The dependency set of j by v p s the set of all tems j n M such that there exsts S M \{j} such that v p (j S v p (j S \{j }. For each such j, we say that j depends on j by p and denote t by j p j, or by j S p j f we want to explctly menton the set S. We denote the dependency set of j by v p by Dep p (j. p or v p may be omtted n any of the above, when t s clear from the context. The relaton s symmetrc (see Appendx A, so we may also use the termnology are dependent and the notaton. Defnton 2.7 The supermodular dependency set of j by v p s the set of all tems j n M such that there exsts S M \{j} such that v p (j S > v p (j S\{j }. Termnology and notatons are the same as n Defnton 2.6, but wth the word supermodular/ly or wth + as a superscrpt. Note that the relaton + s also symmetrc (see Appendx A. Defnton 2.8 The dependency degree of v p s defned as DD(v p def = max j M Dep p (j. The supermodular dependency degree (or smply, the supermodular degree of v p s defned as DD + (v p def = max j M Dep + p (j. The (supermodular dependency degree of an nstance of the welfare maxmzaton problem s the maxmum (supermodular dependency degree among all valuaton functons of the nstance. 3

4 Note that any submodular set functon has supermodular degree 0. Note also that DD + (f DD(f for any set functon f. We also use the followng defnton: Defnton 2.9 Let f be a set functon. The (supermodular dependency graph of f s the followng. There s a vertex for each tem and an undrected edge for each par of (supermodularly dependent tems. 2.3 Representng set functons In the welfare maxmzaton problem, the doman of the valuaton functons s exponental n the number of tems. We recall two possble approaches to cope wth ths. The frst s usng an explct representaton model (specfcally, we recall a hypergraph representaton; see [5], [4], [1] and the second s usng oracles, representng set functons by supportng queres wth respect to them (see for example [3] A hypergraph representaton Every set functon f can be represented n a unque way as a hypergraph n whch the vertces are the tems, vertces and hyperedges have weghts assocated wth them, and the value f(s of a subset S of tems equals the sum of all weghts n the subgraph nduced by the correspondng vertces. We observe that two tems share a hyperedge n the hypergraph representaton f and only f they are dependent, and that sharng a hyperedge of postve weght s a necessary condton but not suffcent for beng supermodularly dependent. 3 A succnct representaton Let f : 2 S R + be a set functon and let s def = S. In general, a succnct representaton of f s any representaton that takes space polynomal n s. For example, the hypergraph representaton s succnct for any set functon wth dependency degree bounded by logs (and, of course, for some other set functons, as well Queres oracles We recall the defntons of value and demand queres oracle for an underlyng set functon. Let f : 2 S R + be a set functon. Defnton 2.10 Value queres are the followng: Input: A subset S S. Output: f(s. Defnton 2.11 Demand queres are the followng: Input: A cost functon c : S R +. Output: A subset S S maxmzng f(s j S c(j. 3 For example, consder the set functon f that has value 0 on the empty set and 1 elsewhere. In ts hypergraph representaton every odd set of tems forms a hyperedge of weght 1 and every even set of tems forms a hyperedge of weght 1. Every two tems share a postve hyperedge, but no two tems are supermodularly dependent, snce f s submodular. 4

5 Note that demand queres are strctly stronger than value queres (see [3]. Defnton 2.12 Queres oracle for a type of queres for a gven set functon can answer queres of the respectve type wth respect to the gven set functon. The noton of an oracle serves as an abstracton for a subroutne that computes an answer to the respectve type of queres for a gven set functon. When we say that an algorthm uses a certan type of oracle, the runnng tme of the algorthm s computed as f each query takes unt tme to answer, regardless of the true runnng tme of the underlyng subroutne. Ths abstracton s most justfed f for the underlyng set functon, answers to the respectve query can ndeed be computed effcently. We remark that gven a succnct hypergraph representaton for a set functon, one can effcently answer value queres, whereas answerng demand queres mght be N P-hard (see for example Theorem 2.4. We also remark that gven a succnct hypergraph representaton, one can construct the correspondng dependency graph and supermodular dependency graph. 2.4 Prelmnary observatons APX-hardness of the welfare maxmzaton problem Proposton 2.1 The welfare maxmzaton problem s APX-hard even f there are only three players who all have the same valuaton functon f, wth DD + (f = 0 and DD(f = 3. Proof: It s known that the queston of whether a 3-regular graph can be legally 3-colored s NP-hard, and that t s APX-hard to maxmze the number of legally colored edges [16]. Gven a 3-regular graph G, consder a set functon f that has G as ts hypergraph representaton, wth all vertces havng value 3 and all edges havng value 1. Assocatng a color wth each player, the allocaton that maxmzes welfare s the one that mnmzes the number of monochromatc edges. The proposton follows Hardness of welfare maxmzaton as a functon of the dependency degree Recall that the maxmum weghted k-set packng problem s the followng: Defnton 2.13 Let G = (V,E,w be a weghted k-unform hypergraph (.e. every hyperedge contans exactly k vertces wth set of vertces V, set of undrected edges E and edge weghts functon w. The maxmum weghted k-set packng problem s to fnd a set of dsjont edges of maxmum weght. Ths problem s known to be NP-hard for any k > 2 and s hard for approxmaton wth guarantee Ω( lnk k, even n the unweghted case, by a result of Hazan, Safra and Schwartz [14]. The best approxmaton guarantee known for t currently (as far as we know s (k +1/2, by an algorthm of Berman [2]. We show the followng: Proposton 2.2 There exsts an approxmaton preservng reducton of the maxmum weghted k-set packng problem to the welfare maxmzaton problem wth dependency degree at most k 1. Proof: The reducton s Reducton 2.1. It s easy to verfy that any feasble soluton for I SP (the nput of Reducton 2.1 has a correspondng feasble soluton for I (the output of Reducton 2.1, wth the same value, and vce versa, 5

6 Reducton 2.1 k-set packng to welfare maxmzaton wth dependency degree at most k 1 Input: An nstance I SP (V,E,w of the maxmum weghted k-set packng problem, wth V = {v 1,...,v V }. Output: An nstance I(P, M, v of the welfare maxmzaton problem wth dependency degree at most k 1. 1: For each vertex v V, create an tem M. 2: For each hyper-edge e = {v e1,...,v ek } E, create a player p e P wth v p (S = w(e for any S such that {e 1,...,e k } S and v p (S = 0 otherwse. as desred An exact algorthm for dependency degree at most 1 Proposton 2.3 The welfare maxmzaton problem wth dependency degree at most 1 admts an exact polynomal tme algorthm. The man dea s to use symmetry of the dependency relaton n order to reduce an nstance of the welfare maxmzaton problem to an nstance of maxmum weghted matchng. The full reducton appears n Appendx C Demand queres and the dependency degree Theorem 2.4 Gven a hypergraph representaton of any set functon wth dependency degree at most 2, demand queres may be answered n polynomal tme (n the sze of the hypergraph representaton. Gven a hypergraph representaton of a set functon wth dependency degree at least 3, demand queres are generally APX-hard to answer (wth respect to the sze of the hypergraph representaton. Proof: Let f be a set functon. If DD(f 2, then each tem depends on at most two other tems. The dependency graph of f s of maxmum degree 2 and hence ts connected components are ether solated vertces, solated paths or solated cycles. On each such component, demand queres may be answered usng dynamc programmng. To show hardness for set functons f wth DD(f 3, we reduce from the problem of maxmum ndependent set n 3-regular graphs. f s represented by the followng hypergraph representaton. Gve each vertex value 3 and each edge value 1. The value of the answer to a demand query n whch the cost of each vertex s 2 s equal to the sze of the maxmum ndependent set (t s worth takng a vertex only f t does not contrbute an edge to the nduced subgraph. Due to APX-hardness of maxmum ndependent set n 3-regular graphs, t s APX-hard to answer demand queres, as well. 2.5 Our man results Wevewthenotonofthesupermodulardegreeandthestudyoftsbascpropertesasanmportant contrbuton of our work. Ths noton s applcable to any set functon, and ts relevance to the welfare maxmzaton problem s demonstrated by the followng theorem. 6

7 Theorem 2.5 The welfare maxmzaton problem wth supermodular degree at most d admts a 1 polynomal tme greedy d+2-approxmaton algorthm. Ths algorthm requres for each valuaton functon a value queres oracle and a supermodular dependency graph. For the dependency degree we have a slghtly better approxmaton guarantee, whch s relevant only when the dependency degree and the supermodular degree are equal: Theorem 2.6 The welfare maxmzaton problem wth dependency degree at most d admts a greedy 1 approxmaton algorthm. Its runnng tme s polynomal n the number of players and tems and n 2 d. Ths algorthm requres for each valuaton functon a value queres oracle and a dependency graph. Proposton 2.2 mples that an mprovement of our results by a multplcatve factor of more than roughly 2 would mprove the current approxmaton guarantee for weghted k-set packng. Addtonally, by a hardness result of Blumrosen and Nsan [3] of Ω(log m/m for algorthms requrng only value queres oracle 4, our bounds are tght up to a multplcatve factor of O(logm, n the sense that general set functons have dependency degree (and supermodular degree at most m Related work The supermodular degree s a complexty measure for set functons, that ranges from 0 (for submodular set functons to m 1 (where m s the number of tems. Our man result s a greedy algorthm for the welfare maxmzaton problem whose approxmaton guarantee ncreases lnearly wth the supermodular degree of the underlyng set functons. Such a lnear ncrease s to be expected, gven the known reducton from set packng to the welfare maxmzaton problem wth sngle mnded bdders, combned wth the dffculty of approxmatng set packng ([14]. As far as we know, the noton of supermodular degree has not appeared n prevous work. However, other related notons dd appear, and n ths secton we dscuss some of them. Perhaps the smplest complexty measure for a set functon s ts support sze, namely, the number of tems that t depends on. Ths measure ranges from 1 to m. Moreover, smple greedy algorthms approxmate the welfare maxmzaton problem wth guarantee equal to ths measure, and known hardness results for the case of sngle mnded bdders apply here as well. Hence the results that one can prove for the complexty measures of support sze and supermodular degree are of a smlar nature. However, as the supermodular degree of a functon s not greater than ts support sze, and moreover, t s often much smaller (the gap beng most dramatc for submodular functons, we vew our results for supermodular degree as sgnfcantly more nformatve than the correspondng results for support sze. Lehmann, Lehmann and Nsan [15] proposed a herarchy of classes of set functons based on notons of complement-freeness, and ntated a study of the complexty of the welfare maxmzaton problem for these classes. For the lower classes (lnear functons, and functons enjoyng the gross substtutes property the welfare maxmzaton problem can be solved n polynomal tme. For 4 Our algorthms requre also dependency graph / supermodular dependency graph. However, n the nstance used by [3], t s trval to construct both, snce all the tems supermodularly depend on each other (for nfntely many values of m. Alternatvely, snce the approxmaton hardness s Ω(log m/m, we may smply modfy the valuatons functons so that every two tems are supermodularly dependent, by addng a postve hyperedge that contans all tems. Detals omtted. 7

8 submodular functons, a constant approxmaton guarantee s possble and value queres suffce for ths [15, 18] (and somewhat better constants are achevable usng demand queres [9]. For XOS (later referred to as fractonally subaddtve n [8] and subaddtve set functons there are approxmaton algorthms wth constant approxmaton guarantees [6, 17, 8], but they requre demand queres (value queres do not suffce [6]. The algorthm gven n [15] for submodular functons s greedy, and the greedy algorthm n the current paper can be vewed as an extenson of the greedy algorthm of [15] to a settng that s not submodular. The classfcaton of [15] does not dstngush between dfferent classes of functons that are not subaddtve, and hence unlke our supermodular degree measure, s applcable only to some classes of set functons, but not to set functons n general. Set functons not lyng n the classfcaton of [15] may have any supermodular degree between 1 and m and our algorthms for maxmzng welfare dstngush among them n the approxmaton guarantees that they provde. Lehmann, Lehmann and Nsan [15] do suggest a way of extendng ther classfcaton to addtonal functons, as follows. A functon f can be called c-submodular, f for every (possbly empty sets S and T and tem x, the margnal value of x wth respect to S T s at most c tmes larger that the margnal value of x wth respect to S. (For submodular functons c = 1. It s shown n [15] that the welfare maxmzaton problem can be approxmated wth guarantee of c + 1 when the set functons are c-submodular. We note however that even functons on two tems need not be c-submodular for any fnte c (f one of the tems has value 0 by tself but postve margnal value together wth the other tem. Submodular functons play a central role n the defnton of the supermodular degree. However, other classes wthn the herarchy of [15] have no specal sgnfcance n ths respect. The supermodular degree does not dstngush between lnear functons and arbtrary submodular functons they both have supermodular degree 0. Functons n the [15] herarchy whch are not submodular may have arbtrarly large supermodular degree. 5 Contzer, Sandholm and Sant [5] ntroduce the representaton of set functons va hypergraphs wth postve and negatve hyperedges (presented n Secton 2.3. See also the work of Chevaleyre, Endrss, Estve and Maudet [4], who defned ndependently a smlar concept. They showed that even f each hyperedge has at most two vertces, the welfare maxmzaton problem s N P-hard. They dd not consder approxmaton algorthms. Abraham, Babaoff, Dughm and Roughgarden [1] consder supermodular functons whch have no negatve hyperedges n ther hypergraph representaton. Among other results, they gve an algorthm approxmatng the welfare maxmzaton problem wthn a value equal to the maxmum cardnalty of any hyperedge (whch may be smaller than the supermodular degree. They prove that obtanng smlar approxmaton guarantees n the presence of negatve hyperedges s NP-hard. Ths serves as an explanaton of why ther model forbds negatve hyperedges. Our results are to some extent n dsagreement wth ths concluson of [1]. Gven a hypergraph representaton of a set functon wth a gven supermodular degree, addng negatve hyperedges cannot ncrease the supermodular degree (n fact, t may cause the supermodular degree to decrease and hence wll not hurt our bounds on the approxmaton guarantees. Ths dscrepancy between our results and those of [1] s explaned by our requrement that set functons are nondecreasng, whereas the hardness of approxmaton results presented n [1] used set functons that are sometmes decreasng. 5 For example, partton the set of tems nto two dsjont sets, A and B. Let f A be a lnear functon that gves value 1 to each tem n A and 0 to each tem n B. Let f B be a lnear functon that gves value 1 to each tem n B and 0 to each tem n A. Let f be defned as f(s = max[f A(S,f B(S]. Ths s an XOS functon (accordng to the classfcaton of [15], but ts supermodular degree s max[ A, B ] 1. 8

9 3 Approxmaton guarantee lnear n supermodular degree In ths secton we prove Theorem 2.5. Our result may be seen as an extenson of awork of Lehmann, Lehmann and Nsan [15], who presented a greedy 2-approxmaton algorthm for submodular set functons. 3.1 The algorthm The algorthm s greedy. In a gven teraton, for every player p and tem j, let D + p (j denote the set of tems not yet allocated that have supermodular dependency wth j wth respect to v p. The algorthm computes the player p and tem j for whch the margnal value for p (gven the tems that p already has of the set j D + p (j s maxmzed, and allocates j D + p (j to p. For a full descrpton of the algorthm, see Algorthm 3.1. Algorthm 3.1 Greedly Approxmate Welfare Maxmzaton wth Guarantee Lnear n Supermodular Degree Input: An nstance I(P, M, v of the welfare maxmzaton problem. A value queres oracle and a supermodular dependency graph for each of the valuaton functons. 1 Output: A soluton wth approxmaton guarantee I. d+2, where d s the supermodular degree of 1: Unallocated M, Approx 2: whle Unallocated do 3: MaxMargnalUtlty 1 4: for all j Unallocated, p P do 5: f v p (j,dep + p (j Unallocated {j M (j p Approx} > MaxMargnalUtlty then 6: MaxMargnalUtlty v p (j,dep + p (j Unallocated {j M (j p Approx} 7: BestAllocaton ({j} (Dep + p (j Unallocated p 8: WnnngPlayer p, AllocatedItem j 9: end f 10: end for 11: Approx Approx BestAllocaton 12: Unallocated Unallocated\(AllocatedItem Dep + WnnngPlayer (AllocatedItem 13: end whle We show Algorthm 3.1 has approxmaton guarantee 1/(d + 2, usng a hybrd argument. Ths wll prove Theorem 2.5. Proof: [of Theorem 2.5] Let OPT be an optmal soluton wth value opt and let APPROX be the output of Algorthm 3.1 wth value approx. For teraton of the loop at lne 2 of Algorthm 3.1, let APPROX be the allocatons made at the frst teratons, let OPT be the allocatons made by OPT for the tems that have not yet been allocated and let HYBRID = APPROX OPT 9

10 be a hybrd soluton. Let HYBRID p and OPT p be the tems allocated n HYBRID and OPT (respectvely to player p P. Note that HYBRID 0 = OPT and HYBRID t = APPROX, where t s the total number of teratons. We prove the followng lemma: Lemma 3.1 Let be an teraton of the loop at lne 2 of Algorthm 3.1 and let p be the player to whom tems are allocated at teraton. Then, (d+2(v p (APPROX p v p (APPROX p 1 n ( vp (OPT p 1 APPROXp 1 v p(opt p APPROX p. p=1 That s, the value lost by any teraton s bounded by d + 2 tmes the value ganed by the same teraton. Proof: Let x = v p (APPROX p v p (APPROX p 1. Roughly speakng, we prove that: 1. For an tem allocated to some p p n OPT, the loss to the value of p for not gettng the tem s at most x. 2. Havng receved tems n the current teraton, the loss n margnal value of future tems gven to p s at most x. The frst contrbutes to the damage up to ( x, snce at most tems are allocated at each teraton. The second contrbutes up to another x, and for any other player HYBRID 1 = HYBRID. We prove the lemma formally. Let j 1,...j d be the tems allocated at teraton and let P be the set of players p p such that at least one of the tems j 1,...,j d s allocated to p n OPT 1. Let p P and let ĵ 1,...,ĵ d {j 1,...,j d } be all the tems of j 1,...,j d, allocated to 10

11 p by OPT 1. Then, v p (OPT p 1 APPROX p 1 v p (OPT p APPROX p ( = v p {ĵ 1,...ĵ d } OPT p APPROX p 1 d d = v p (ĵ k {ĵ k } OPT p APPROX p k=1 d d d d max k=1 v p d max k=1 v p d max k=1 v p k =k+1 (ĵ k d k =k+1 (ĵ k (( d ( ĵ k 1 {ĵ k } OPT p APPROX p k =k+1 (( d k =k+1 {ĵ k } OPT p {ĵ k } OPT p 1 Dep + p (ĵ k Dep + p (ĵ k APPROX p 1 APPROXp 1 d d max v p ĵ k=1 k Dep + p (ĵ k \ APPROX p 1 p P APPROXp ( d v p j 1,...,j d APPROX p 1 ( = d v p (APPROX p v p (APPROX p 1 where, the frst equalty follows by defntons and by observng that for any player p p, APPROX p 1 = APPROXp ; the second by defntons. The frst nequalty s trval; the second follows by Defnton 2.7; the thrd and fourth by monotoncty; the ffth by lne 5 of Algorthm 3.1. The last equalty follows by defntons. Snce there are only d tems allocated, and snce for any player p / P {p }, HYBRID p = HYBRIDp 1, we get, ((v p (APPROX p v p (APPROX p 1 (v p (OPT p 1 APPROXp 1 v p(opt p APPROX p p P\{p } For player p, we have by monotoncty v p (HYBRID p v p (HYBRID p 1. Hence, 1. (1 v p (OPT p 1 APPROXp 1 +v p (APPROXp 1 v p (OPTp APPROX p +v p (APPROX p and then, v p (OPT p 1 APPROXp 1 v p (OPTp APPROX p v p (APPROX p v p (APPROX p 1. Ths and (1 prove Lemma 3.1. We use Lemma 3.1 to complete the proof of Theorem 2.5. Let x be the proft of Algorthm

12 at teraton. Then, opt = = = n v p (OPT p 0 APPROXp 0 p=1 n (v p (OPT p 0 APPROXp 0 v p(opt p t APPROX p t p=1 n t 1 ( vp (OPT p APPROX p v p(opt p +1 APPROXp +1 p=1 =0 (d+2 t x = (d+2 approx =1 Ths proves Theorem A tght example The followng example shows Algorthm 3.1 may return a soluton wth value arbtrarly close to, whch matches the upper bound we proved n Theorem d+2 Example 3.2 Let I(P,M,v be an nstance of the welfare maxmzaton problem wth players P = {1,2}, tems M = {j,j 1,...,j d,j } and valuaton functons v 1 as n the hypergraph representaton shown n Fgure 1 below and v 2 (S = S \{j }, for any S M. It s easy to observe the optmal soluton gves all the tems except j to player 2 and j to player 1. The value of ths soluton s d+2. On the other hand, Algorthm 3.1 gves all the tems except j to player 1, and gves j to an arbtrary player. Ths soluton has value of only 1+d ε, whch tends to 1 as ε decreases. Fgure 1: Valuaton functon of player 1 4 Approxmaton guarantee lnear n dependency degree We dscuss two possble alternatves for provng Theorem 2.6. One s adaptng Algorthm 3.1, as we brefly dscuss n ths secton and the other s presented n Appendx B. We sketch a possble 12

13 adaptaton of Algorthm 3.1. Recall that Algorthm 3.1 does the followng n each teraton: for each player and tem, t calculates the margnal value of the tem and all ts unallocated supermodular dependences wth respect to the allocated tems of the player. A player and tems wth maxmum value are selected. The modfcaton here s twofold. Frstly, for each player and tem, one consders not only the subset of all unallocated dependences of the tem, but any possble subset of unallocated dependences. Secondly, one does not consder the margnal value of an tem together wth ts dependences wth respect to the prevously allocated tems, but only of the tem, wth respect to the subset of dependences under consderaton and the prevously allocated tems, together. It can be shown that the damage for any player, caused by an allocaton of a sngle tem to another player, s bounded by the proft of the teraton damagng t, and also (unlke the case of supermodular dependency that the player gettng the allocaton has no damage at all n future teratons. The other alternatve, fully presented n Appendx B, s desgnng a dfferent greedy algorthm. Ths algorthm has the somewhat surprsng property that when consderng whch tems to add to a player, t completely gnores the tems that the player already has (despte the fact that these tems determne the margnal values for new tems. 5 Acknowledgements Ths research was supported by The Israel Scence Foundaton (grant No. 621/12. We thank Inbal Talgam-Cohen for her useful comments on an earler verson of ths manuscrpt. References [1] I. Abraham, M. Babaoff, S. Dughm, and T. Roughgarden. Combnatoral auctons wth restrcted complements. In EC, [2] P. Berman. A d/2 approxmaton for maxmum weght ndependent set n d-claw free graphs. Nordc Journal of Computng, 7: , Prelmnary verson n SWAT 00. [3] L. Blumrosen and N. Nsan. On the computatonal power of demand queres. SIAM Journal on Computng, 39: , [4] Y. Chevaleyre, U. Endrss, S. Estve, and N. Maudet. Multagent resource allocaton n k- addtve domans: preference representaton and complexty. Annals of Operatons Research, 163:49 62, [5] V. Contzer, T. Sandholm, and P. Sant. Combnatoral auctons wth k-wse dependent valuatons. In AAAI, pages , [6] S. Dobznsk, N. Nsan, and M. Schapra. Approxmaton algorthms for combnatoral auctons wth complement-free bdders. Mathematcs of Operatons Research, 35:1 13, Prelmnary verson n STOC 05. [7] J. Edmonds. Maxmum matchng and a polyhedron wth 0, 1-vertces. Journal of Research of the Natonal Bureau of Standards, 69: ,

14 [8] U. Fege. On maxmzng welfare when utlty functons are subaddtve. SIAM Journal on Computng, 39: , Prelmnary verson n STOC 06. [9] U. Fege and J. Vondrák. Approxmaton algorthms for allocaton problems: Improvng the factor of 1 1/e. In FOCS, pages , [10] M. L. Fsher, G. L. Nemhauser, and L. A. Wolsey. An analyss of approxmatons for maxmzng submodular set functons - II. Mathematcal Programmng Study, 8:73 87, [11] H. N. Gabow. Implementaton of algorthms for maxmum matchng on nonbpartte graphs. PhD thess, Stanford Unversty, [12] Z. Gall. Effcent algorthms for fndng maxmal matchng n graphs. Techncal report, Columba Unversty, New York, [13] Z. Gall, S. Mcal, and H. N. Gabow. An O(EV logv algorthm for fndng a maxmal weghted matchng n general graphs. SIAM Journal on Computng, 15: , [14] E. Hazan, S. Safra, and O. Schwartz. On the complexty of approxmatng k-set packng. Computatonal Complexty, 15:20 39, [15] B. Lehmann, D. J. Lehmann, and N. Nsan. Combnatoral auctons wth decreasng margnal utltes. Games and Economc Behavor, 55: , Prelmnary verson n EC [16] E. Petrank. The hardness of approxmaton: Gap locaton. Computatonal Complexty, 4: , [17] M. Schapra S. Dobznsk. An mproved approxmaton algorthm for combnatoral auctons wth submodular bdders. In SODA, pages , [18] J. Vondrák. Optmal approxmaton for the submodular welfare problem n the value oracle model. In STOC, pages 67 74, A Symmetry of dependency relatons Lemma A.1 (Symmetry Let p P and let j 1,j 2 M be such that j 1 p j 2. Then j 2 p j 1. In other words, the relaton p s symmetrc. Note that the same s true for the relaton + and that the proof s exactly the same. S Proof: Let S be such that j 1 j 2. We show that From Defnton 2.5, on one hand, and on the other hand, S \ {j 2 } {j 1 } j 2 j 1. v({j 1 } S = v(j 1 S+v(j 2 S \{j 2 }+v(s \{j 2 }, v({j 1 } S = v(j 2 S \{j 2 } {j 1 }+v(j 1 S \{j 2 }+v(s \{j 2 }. 14

15 By subtractng v(s \{j 2 }, we get: v(j 1 S+v(j 2 S \{j 2 } = v(j 2 S \{j 2 } {j 1 }+v(j 1 S \{j 2 }. S Snce j 1 j 2 means v(j 1 S v(j 1 S \{j 2 }, we get v(j 2 S \{j 2 } v(j 2 S \{j 2 } {j 1 }. S \ {j 2 } {j 1 } The latter s exactly the defnton of j 2 j 1, as desred. B A greedy 1 at most d In ths appendx we prove Theorem approxmaton algorthm for dependency degree B.1 The algorthm The algorthm s Algorthm B.1. Algorthm B.1 Greedly Approxmate Welfare Maxmzaton wth Guarantee Lnear n Dependency Degree Input: An nstance I(P, M, v of the welfare maxmzaton problem. A value queres oracle and a dependency graph for each of the valuaton functons. 1 Output: A soluton wth approxmaton guarantee, where d s the dependency degree of I. 1: Unallocated M, Approx 2: whle Unallocated do 3: MaxMargnalUtlty 1 4: for all j Unallocated, p P, S (Dep p (j Unallocated do 5: f v p (j S > MaxMargnalUtlty then 6: MaxMargnalUtlty v p (j S 7: BestAllocaton ({j} S p 8: WnnngPlayer p, AllocatedItem j, AllocatedOptmalDependences S 9: end f 10: end for 11: Approx Approx BestAllocaton 12: U nallocated U nallocated\(allocateditem AllocatedOptmalDependences 13: Unallocated Unallocated \ Dep WnnngPlayer (AllocatedItem {Dscard also unallocated dependences of j} 14: end whle Intutvely, Algorthm B.1 promses at each teraton that the most possbly contrbutng tem wll have ts full contrbuton n the approxmated soluton. Thus, any mslocated tem n an optmal soluton cannot damage t n more than the beneft of the teraton damagng t. We conclude the approxmaton guarantee by observng no more than tems are allocated at each teraton. 15

16 Few remarks are n place. Remark B.1 Algorthm B.1 does not look at all at the value of already allocated tems (nether at new nspected tems relatvely to them nor at the whole sub-allocaton. Note that an approach lookng only at the margnal value wth respect to already allocated tems wthout lookng on forward dependences does not work. 6 Remark B.2 Algorthm B.1 dscards unallocated dependences (at lne 13. Ths s to ensure any selected tem wll have ts nspected margnal value. In other words, we ensure that the only dependences of an tem at the rest of the soluton (.e. the unallocated part wll be ts optmal dependences, as nspected at lnes Of course, because of monotoncty, we may add the dscarded tems to any player we wsh (for example to the player we allocated to the rest or each tem to ts best possblty, n any order. The tght example we wll show s tght also for any of these possbltes. Proof: [of Theorem 2.6] Let OPT be an optmal soluton wth value opt and let APPROX be an output of Algorthm B.1 wth value approx. Let t be the number of teratons of the whle loop at lne 2 of the run created APPROX. We layer opt and approx by wrtng both of them by teratons of Algorthm B.1. opt = approx = t v popt(,k(j,k OPT popt(,k \ =1 k=1 1 t v papp((j,k APPROX papp( \ =1 k=1 =1k =1 1 =1k =1 k {j,k }\ {j,k } k =1 k {j,k }\ {j,k } k =1 where: j,k s the k th tem allocated at teraton, where j,1 s the fnal tem assgned at lne 7 of ths teraton, and the rest are ordered arbtrarly. p opt (,k s the player to whom the k th tem of teraton s allocated n OPT. p app ( s the player to whom all the tems of teraton are allocated n APPROX (all tems of any teraton of Algorthm B.1 are allocated to the same player. Note that for smplcty, equatons are wrtten assumng exactly d + 1 tems are allocated at each teraton. The proof s correct also wthout ths assumpton. Let [1..t] and let Unallocated be the tems of Unallocated at lne 4 of Algorthm B.1 at teraton. Then, snce M = n (OPT p = n (APPROX p, we have: Unallocated = p=1 ( n OPT p \ p=1 1 =1k =1 p=1 {j,k } = ( n APPROX p \ p=1 1 =1k =1 {j,k } 6 For example, we may have two tems and two players, where the frst player has set functon f(s = S and the second has set functon gvng to both tems together and 0 otherwse. An algorthm that looks only backward wll allocate both tems to player 1 and gan value 2 where an optmal soluton has value. 16

17 Therefore, for all [1..t], k [1..], ( n OPT p \ p=1 1 =1k =1 k {j,k }\ {j,k } Unallocated. k =1 Then, by lnes 4-10 and 13 of Algorthm B.1, for all [1..t], k [1..d + 1], v papp((j,1 {j,k }\{j,1} v popt(,k(j,k OPT 1 popt(,k\ {j,k }\ k {j,k }. APPROX papp(\ 1 =1k =1 Therefore and by nvokng (2 and (2 together wth monotoncty, ( approx = t ( v papp((j,k APPROX papp( \ =1 ( k t v papp((j,1 APPROX papp( \ =1 t v popt(,k(j,k OPT popt(,k \ =1 = opt k 1 =1k =1 1 =1k =1 1 =1k =1 =1k =1 k {j,k }\ {j,k } k =1 {j,k }\{j,1} k {j,k }\ {j,k } It s easy to see the runnng tme of Algorthm B.1 s polynomal n M, P and 2 c. Ths proves Theorem 2.6. k =1 k =1 B.1.1 An example justfyng dscardng tems We demonstrate the necessty of dscardng unused dependences as n lne 13 of Algorthm B.1. Note that another approach s to look also on already allocated tems, as descrbed n Secton 4. Example B.1 Let P = 2. Let the set functons be as follows: The valuaton functon of player 1 wll be as n the hypergraph representaton n Fgure 2. The valuaton functon of player 2 wll be v 2 (S = S. Intutvely, the dea s to cause Algorthm B.1 to run a margnal value of an already allocated tem, when tryng to gan a maxmal margnal value for another tem. The mddle edge wth value 1 does so, wthout breakng monotoncty. We analyze the approxmaton guarantee for ths nstance for Algorthm B.1 wth lne 13 omtted (.e. wthout dscardng unused dependences. On the frst teraton the algorthm allocates ether j 1 or j 2 to player 1 wth ther optmal dependences. Assume wthout loss of generalty t s j 1. The optmal dependences of j 1 are j 1,1,...,j 1,d 1 and the margnal value of j 1 wth these dependences s 1+(d 1 ε. On the second teraton, the algorthm allocates j 2 to player 1, together wth ts optmal dependences (that has not been allocated yet j 2,1,...,j 2,d 1. The margnal value of j 2 wth respect to these dependences s also 1 + (d 1 ε. But now, the margnal value of j 1 wth respect to the rest of the tems allocated to player 1 (whch had not been allocated yet when t was allocated s only (d 1 ε. The algorthm termnates after the second teraton wth socal welfare 17

18 Fgure 2: Valuaton functon of player 1 1+2(d 1 ε. On the other hand, allocatng all the tems to player 2 results n a socal welfare of 2d. For small enough ε the approxmaton guarantee s arbtrarly close to 1/2d whch s much worse than the approxmaton guarantee of 1/(d + 1 of Algorthm B.1. Note that Algorthm B.1 (the unmodfed verson does much better for ths nstance. It also chooses frstly j 1 (wthout loss of generalty wth ts optmal dependences, but then does not nspect anymore any of ts dependences, ncludng j 2. Therefore, assumng ε s small, t allocates j 2,1,...,j 2,d 1 to player 2 and gans for them a value of d 1 n addton to the margnal value t ndeed ganed for j 1. Thus, the approxmated soluton s total value Algorthm B.1 gans for ths nput s 1+(d 1 ε+(d 1 = d+(d 1 ε, whch expresses an approxmaton guarantee of slghtly more than 1 2. Note that allocatng j 2 to any of the players wll do no harm (and allocatng t to player 2 wll even slghtly help; just renspectng t does the harm. B.2 A tght example We now show the analyss of Algorthm B.1 s tght. Example B.2 Let P = 2 and let M = m (d + 1 for some m N. We set an arbtrary orderng on the tems and defne m subsets of tems S 1,...S m ; the frst set wll be the frst tems, the second one wll be the next tems and so on. Let v be the followng set functons for any S M: v 1 (S = [1..m ], S S (1+ε (meanng, (1+ε tmes the number of subsets S that are subsets of S. v 2 (S = S. It s easy to see the margnal value of any tem s maxmzed, when t s gven to player 1 wth all ts dependences. Moreover, snce at ths way only whole subsets are allocated, ths s the stuaton 18

19 at any teraton of Algorthm B.1. Therefore, algorthm B.1 allocates all the tems to player 1 and gans total value of m (1+ε. In contrast, the optmal soluton s to allocate all the tems to player 2 and to gan total value of m. Thus, the approxmaton guarantee for ths nstance s close as we wth to m/m = 1/(, and the analyss of Theorem 2.6 s ndeed tght. C An exact algorthm for dependency degree at most 1 In ths appendx, we present a full reducton (Reducton C.1 of the welfare maxmzaton problem wth dependency degree at most 1 to the maxmum weghted matchng problem. Reducton C.1 Input: An nstance I(P, M, v of the welfare maxmzaton problem. Output: An nstance I M (V,E,w of the maxmum weghted matchng problem, wth set of vertces V, set of undrected edges E and edge weghts functon w, such that each tem j M s represented by a correspondng vertex v j V. Reducton: For each tem j, we have a vertex u j. For each player p, we have the followng. For each tem j wth Dep p (j =, we have an auxlary vertex u p j and an edge (u j,u p j wth weght v p ({j}, representng the possblty of allocatng j to p. For each par of tems j,j, such that j p j, we have a sngle auxlary vertex u p j,j and three edges: (u j,u p j,j wth weght v p ({j}, representng the possblty of allocatng j to p wthout allocatng j to p; (u j,u p j,j wth weght v p ({j }, representng the possblty of allocatng j to p wthout allocatng j to p; (u j,u j wth weght v p ({j,j }, representng the possblty of allocatng both j and j to p. Note that both j and j have no other dependences, snce the dependency degree s at most 1 and by Lemma A.1. Note also that multedges may be resolved, by choosng, wthout loss of generalty, one edge wth maxmum weght for each par of vertces. The followng observaton s straghtforward: Observaton C.1 Reducton C.1 s polynomal tme computable. Every feasble soluton for I M nduces a feasble soluton for I wth the same value, that may be computed n polynomal tme. Every feasble soluton for I, nduces a feasble soluton for I M wth at least the same value. Therefore, we may use Reducton C.1 together wth any exact polynomal tme maxmum weghted matchng algorthm (see for example [7, 11, 12, 13] to derve an exact polynomal tme algorthm for the welfare maxmzaton problem wth dependency degree at most 1. Remark C.1 Note that Reducton C.1 does not work for c > 1. Ths s snce we may have a player p wth tems j 1,j 2,j 3, such that j 1 j 2 j 3 (.e. a chan. At ths case, j 1,j 2 and j 3 may be allocated together to p, although ths allocaton does not nduce a feasble matchng. 19