Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions

Transcription

1 Inapproximability Rsults for Combinatorial Auctions with Submodular Utility Functions Subhash Khot Richard J. Lipton Evanglos Markakis Aranyak Mhta Abstract W considr th following allocation problm arising in th stting of combinatorial auctions: a st of goods is to b allocatd to a st of playrs so as to maximiz th sum of th utilitis of th playrs (i.., th social wlfar). In th cas whn th utility of ach playr is a monoton submodular function, w prov that thr is no polynomial tim approximation algorithm which approximats th maximum social wlfar by a factor bttr than 1 1/ 0.632, unlss P= NP. Our rsult is basd on a rduction from a multi-provr proof systm for MAX-3-COLORING. 1 Introduction A larg volum of transactions is nowadays conductd via auctions, including auction srvics on th intrnt (.g., Bay) as wll as FCC auctions of spctrum licncs. Rcntly, thr has bn a lot of intrst in auctions with complx bidding and allocation possibilitis that can captur various dpndncis btwn a larg numbr of itms bing sold. A vry gnral modl which can xprss such complx scnarios is that of combinatorial auctions. In a combinatorial auction, a st of goods is to b allocatd to a st of playrs. A utility function is associatd with ach playr spcifying th happinss of th playr for ach subst of th goods. On natural objctiv for th auctionr is to maximiz th conomic fficincy of th auction, which is th sum of th utilitis of all th playrs. Formally, th allocation problm is dfind as follows: W hav a st M of m indivisibl goods and n playrs. Playr i has a monoton utility function v i : 2 M R. W wish to find a partition (S 1,..., S n ) of th st of goods among th n playrs that maximizs th total utility or social wlfar, i v i(s i ). Such an allocation is calld an optimal allocation. Not that th actual amount of mony paid by a playr during th auction is xtranous to th problm of maximizing th social wlfar, and w do not considr it hr. W ar intrstd in th computational complxity of th allocation problm. W would lik an algorithm which runs in tim polynomial in n and m. Howvr, on can s that th input rprsntation is itslf xponntial in m for gnral utility functions. Evn if th utility functions hav a succinct rprsntation (polynomial in n and m), th allocation problm may b NP-hard [12, 1]. In th absnc of a succinct rprsntation, it is typically assumd Gorgia Institut of Tchnology, Atlanta GA 30332, USA, {khot, vanglis, aranyak}@cc.gatch.du Gorgia Institut of Tchnology, Atlanta GA 30332, USA and Tlcordia Rsarch, Morristown NJ 07960, USA, rjl@cc.gatch.du 1

2 that th auctionr has oracl accss to th playrs utilitis and that h can ask quris to th playrs. Thr ar 2 typs of quris that hav bn considrd. In a valu qury th auctionr spcifis a subst S M and asks playr i for th valu v i (S). In a dmand qury, th auctionr prsnts a st of prics for th goods and asks a playr for th st S of goods that maximizs his profit (which is his utility for S minus th sum of th prics of th goods in S). Not that if w hav a succinct rprsntation of th utility functions thn w can always simulat valu quris. Evn with quris th problm rmains hard. Hnc w ar intrstd in approximation algorithms and inapproximability rsults. A natural class of utility functions that has bn studid xtnsivly in th litratur is th class of submodular functions. A function v is submodular if for any 2 sts of goods S T, th marginal contribution of a good x T, is biggr whn addd to S than whn addd to T, i.., v(s x) v(s) v(t x) v(t ). Submodularity can b sn as th discrt analog of concavity and ariss naturally in conomic sttings sinc it capturs th proprty that marginal utilitis ar dcrasing as w allocat mor goods to a playr. It is known that th class of submodular utility functions contains th functions with th Gross Substituts proprty [10], and also that submodular functions ar complmnt-fr. Prvious Work For gnral utility functions, th allocation problm is NP-hard. Approximation algorithms hav bn obtaind that achiv factors of O( 1 m ) ([13, 5], using dmand quris) and O( log m m ) ([11], using valu quris). It has also bn shown that w cannot hav polynomial tim algorithms with a factor bttr than O( log m m ) ([5], using valu quris) or bttr than O( 1 ) m 1/2 ɛ ([13, 18], vn for singl mindd biddrs). Evn if w allow dmand quris, xponntial communication is rquird to achiv any approximation guarant bttr than O( ) [15]. 1 m 1/2 ɛ For singl-mindd biddrs, as wll as for othr classs of utility functions, approximation algorithms hav bn obtaind, among othrs, in [2, 4, 13]. For mor rsults on th allocation problm with gnral utilitis, s [6]. For th class of submodular utility functions, th allocation problm is still NP-hard. Th following positiv rsults ar known: In [12] it was shown that a simpl grdy algorithm using valu quris achivs an approximation ratio of 1/2. An improvd ratio of 1 1/ was obtaind in [1] for a spcial cas of submodular functions, th class of additiv valuations with budgt constraints. Vry rcntly, approximation algorithms with ratio 1 1/ wr obtaind in [7, 8] using dmand quris. As for ngativ rsults, it was shown in [15] that an xponntial amount of communication is ndd to achiv an approximation ratio bttr than 1 O( 1 m ). In [7] it was shown that thr cannot b any polynomial tim algorithm in th succinct rprsntation or th valu qury modl with a ratio bttr than 50/51, unlss P= NP. Our Rsult W show that thr is no polynomial tim approximation algorithm for th allocation problm with monoton submodular utility functions achiving a ratio bttr than 1 1/, unlss P= NP. Our rsult is tru in th succinct rprsntation modl, and hnc also in th valu qury modl. Th rsult dos not hold if th algorithm is allowd to us dmand quris. A hardnss rsult of 1 1/ for th class XOS (which strictly contains th class of submodular

3 Algorithms Hardnss Valu Quris 1/2 [12] 1 1/ Dmand Quris 1 1/ [8] 1 O(1/m) [15] Tabl 1: Approximability rsults for submodular utilitis functions) is obtaind in [7] by a gadgt rduction from th maximum k-covrag problm. For a dfinition of th class XOS, s [12]. Similar rductions do not sm to work for submodular functions. Instad w provid a rduction from multi-provr proof systms for MAX-3-COLORING. Our rsult is basd on th rduction of Fig [9] for th hardnss of st-covr and maximum k-covrag. Th rsults of [9] us a rduction from a multi-provr proof systm for MAX-3-SAT. This is not sufficint to giv a hardnss rsult for th allocation problm, as xplaind in Sction 3. Instad, w us a proof systm for MAX-3-COLORING. W thn dfin an instanc of th allocation problm and show that th nw proof systm nabls all playrs to achiv maximum possibl utility in th ys cas, and nsur that in th no cas, playrs achiv only (1 1/) of th maximum utility on th avrag. Th crucial proprty of th nw proof systm is that whn a graph is 3-colorabl, thr ar in fact many diffrnt proofs (i.., colorings) that mak th vrifir accpt. This would not b tru if w start with a proof systm for MAX-3-SAT. By introducing a corrspondnc btwn colorings and playrs of th allocation instanc, w obtain th dsird rsult. Th currnt stat of th art for th allocation problm with submodular utilitis, including our rsult, is summarizd in Tabl 1. W not that w do not addrss th qustion of obtaining truthful mchanisms for th allocation problm. For som classs of functions, incntiv compatibl mchanisms hav bn obtaind that also achiv rasonabl approximations to th allocation problm (.g. [13, 2, 4]). For submodular utilitis, th only truthful mchanism known is obtaind in [7]. This achivs an O( 1 m )-approximation. Obtaining a truthful mchanism with a bttr approximation guarant sms to b a challnging opn problm. Th rst of th papr is organizd as follows: In th nxt sction w dfin th modl formally and introduc som notation. In Sction 3 w prsnt a wakr hardnss rsult of 3/4 using a 2-provr proof systm to illustrat th idas in our proof. In Sction 4 w prsnt th hardnss of 1 1/ basd on th k-provr proof systm of [9]. 2 Modl, Dfinitions and Notation W assum w hav a st of playrs N = {1,..., n} and a st of goods M = {1,..., m} to b allocatd to th playrs. Each playr has a utility function v i, whr for a st S M, v i (S) is th utility that playr i drivs if h obtains th st S. W mak th standard assumptions that v i is monoton and that v i ( ) = 0. Dfinition 1 A function v : 2 M R is submodular if for any sts S T and any x M\T : v(s {x}) v(s) v(t {x}) v(t ) An quivalnt dfinition of submodular functions is that for any sts S, T : v(s T )+v(s T ) v(s) + v(t ).

4 An allocation of M is a partition of th goods (S 1,..., S n ) such that i S i = M and S i S j =. Th allocation problm w will considr is: Th allocation problm with submodular utilitis: Givn a monoton, submodular utility function v i for vry playr i, find an allocation of th goods (S 1,..., S n ) that maximizs i v i(s i ). To clarify how th input is accssd, w assum that ithr th utility functions hav a succinct rprsntation 1, or that th auctionr can ask valu quris to th playrs. In a valu qury, th auctionr spcifis a subst S to a playr i and th playr rsponds with v i (S). In this cas th auctionr is allowd to ask at most a polynomial numbr of valu quris. Sinc th allocation problm is NP-hard, w ar intrstd in polynomial tim approximation algorithms or hardnss of approximation rsults: an algorithm achivs an approximation ratio of α 1 if for vry instanc of th problm, th algorithm rturns an allocation with social wlfar at last α tims th optimal social wlfar. 3 A Hardnss of 3/4 W first prsnt a hardnss rsult of 3/4. Th rduction of this sction is basd on a 2-provr proof systm for MAX-3-COLORING, which is analogous to th proof systm of [14] for MAX- 3-SAT. W rmark that this proof is providd hr only to illustrat th main idas of our rsult and to giv som intuition. In th nxt Sction w will s that by moving to a k-provr proof systm w can obtain a hardnss of 1 1/. In th MAX-3-COLORING problm, w ar givn a graph G and w ar askd to color th vrtics of G with 3 diffrnt colors so as to maximiz th numbr of proprly colord dgs, whr an dg is proprly colord if its vrtics rciv diffrnt colors. Givn a graph G, lt OP T (G) dnot th maximum fraction of dgs that can b proprly colord by any 3-coloring of th vrtics. W will start with a gap vrsion of MAX-3-COLORING: Givn a constant c, w dnot by GAP-MAX-3-COLORING(c) th promis problm in which th ys instancs ar th graphs with OP T (G) = 1 and th no instancs ar graphs with OP T (G) c. By th PCP thorm [3], and by [16], w know: Proposition 1 Thr is a constant c < 1 such that GAP-MAX-3-COLORING(c) is NP-hard, i.., it is NP-hard to distinguish whthr YES Cas: OP T (G) = 1, and NO Cas: OP T (G) c. Lt G b an instanc of GAP-MAX-3-COLORING(c). Th first stp in our proof is a rduction to th Labl Covr problm. A labl covr instanc L consists of a graph G, a st of labls Λ and a binary rlation π Λ Λ for vry dg. Th rlation π can b sn as a constraint on th labls of th vrtics of. An assignmnt of on labl to ach vrtx is calld a labling. Givn a labling, w will say that th constraint of an dg = (u, v) is satisfid if 1 By this w man a rprsntation of siz polynomial in n and m, such that givn S and i, th auctionr can comput v i(s) in tim polynomial in th siz of th rprsntation. For xampl, additiv valuations with budgt limits [12] hav a succinct rprsntation.

5 (l(u), l(v)) π, whr l(u), l(v) ar th labls of u, v rspctivly. Th goal is to find a labling of th vrtics that satisfis th maximum fraction of dgs of G, dnotd by OP T (L). Th instanc L producd from G is th following: G has on vrtx for vry dg (u, v) of G. Th vrtics (u 1, v 1 ) and (u 2, v 2 ) of G ar adjacnt if and only if th dgs (u 1, v 1 ) and (u 2, v 2 ) hav on common vrtx in G. Each vrtx (u, v) of G has 6 labls corrsponding to th 6 diffrnt propr colorings of (u, v) using 3 colors. For an dg btwn (u 1, v 1 ) and (u 2, v 2 ) in G, th corrsponding constraint is satisfid if th labls of (u 1, v 1 ) and (u 2, v 2 ) assign th sam color to thir common vrtx. From Proposition 1 it follows asily that: Proposition 2 It is NP-hard to distinguish btwn: YES Cas: OP T (L) = 1, and NO Cas: OP T (L) c, for som constant c < 1 W will say that 2 lablings L 1, L 2 ar disjoint if vry vrtx of G rcivs a diffrnt labl in L 1 and L 2. Not that in th YES cas, thr ar in fact 6 disjoint lablings that satisfy all th constraints. Th Labl Covr instanc L is ssntially a dscription of a 2-provr 1-round proof systm for MAX-3-COLORING with compltnss paramtr qual to 1 and soundnss paramtr qual to c (s [9, 14] for mor on ths proof systms). Givn L, w will now dfin a nw labl covr instanc L, th hardnss of which will imply hardnss of th allocation problm. Going from instanc L to L is quivalnt to applying th paralll rptition thorm of Raz [17] to th 2-provr proof systm for MAX-3-COLORING. W will dnot by H th graph in th nw labl covr instanc L. A vrtx of H is now an ordrd tupl of s vrtics of G, i.., it is an ordrd tupl of s dgs of G, whr s is a constant to b dtrmind latr. W will rfr to th vrtics of H as nods to distinguish thm from th vrtics of G. For 2 nods of H, u = ( 1,..., s ) and v = ( 1,..., s), thr is an dg btwn u and v if and only if for vry i [s], th dgs i and i hav xactly on common vrtx (whr [s] = {1,..., s}). W will rfr to ths s common vrtics as th shard vrtics of u and v. Th st of labls of a nod v = ( 1,..., s ) is th st of 6 s propr colorings of its dgs (Λ = [6 s ]). Th constraints can b dfind analogously to th constraints in L. In particular, for an dg = (u, v) of H, a labling satisfis th constraint of dg if th labls of u and v induc th sam coloring of thir shard vrtics. By Proposition 2 and Raz s paralll rptition thorm [17], w can show that: Proposition 3 It is NP-hard to distinguish btwn: YES Cas: OP T (L ) = 1, and NO Cas: OP T (L ) 2 γs, for som constant γ > 0. Rmark 1 In fact, in th YES cas, thr ar 6 s disjoint lablings that satisfy all th constraints. This proprty will b usd crucially in th rmaining sction. Th known rductions from GAP-MAX-3-SAT to labl covr, implicit in [9, 14], ar not sufficint to guarant that thr is mor than on labling satisfying all th dgs. This was our motivation for using GAP- MAX-3-COLORING instad.

6 Th final stp of th proof is to dfin an instanc of th allocation problm from L. For that w nd th following dfinition: Dfinition 2 A Partition Systm P (U, r, h, t) consists of a univrs U of r lmnts, and t pairs of sts (A 1, Ā1),...(A t, Āt), (A i U) with th proprty that any collction of h h sts without a complmntary pair A i, Āi covrs at most (1 1/2 h )r lmnts. If U = {0, 1} t, w can construct a partition systm P (U, r, h, t) with r = 2 h and h = t. For i = 1,..., t th pair (A i, Āi) will b th partition of U according to th valu of ach lmnt in th i-th coordinat. In this cas th sts A i, Āi hav cardinality r/2. For vry dg in th labl covr instanc L, w construct a partition systm P (U, r, h, t = h = 3 s ) on a sparat subunivrs U as dscribd abov. Call th partitions (A 1, Ā 1 ),..., (A t, Ā t ). Rcall that for vry dg = (u, v), thr ar 3 s diffrnt colorings of th s shard vrtics of u and v. Assuming som arbitrary ordring of ths colorings, w will say that th pair (A i, Ā i ) of P corrsponds to th ith coloring of th shard vrtics. W dfin a st systm on th whol univrs U. For vry nod v and vry labl i, w hav a st S v,i. For vry dg incidnt on v, S v,i contains on st from vry partition systm P, as follows. Considr an dg = (v, w). Thn A j contributs to all th sts S v,i such that labl i in nod v inducs th jth coloring of th shard vrtics btwn v and w. Similarly Ā j contributs to all th S w,i such that labl i in nod w givs th jth coloring to th shard vrtics (th choic of assigning A j to th S v,i s and Ā j to th S w,i s is mad arbitrarily for ach dg (v, w)). Thus S v,i = (v,w) E B (v,w) j whr E is th st of dgs of H, B (v,w) j is on of A (v,w) j or Āj (v,w), and j is th coloring that labl i givs to th shard vrtics of (v, w). W ar now rady to dfin our instanc I of th allocation problm. Thr ar n = 6 s playrs, all having th sam utility function. Th goods ar th sts S v,i for ach nod v and labl i. If a playr is allocatd a collction of goods S v1,i 1...S vl,i l, thn his utility is l S vj,i j j=1 It is asy to vrify that this is a monoton and submodular utility function. Lt OP T (I) b th optimal solution to th instanc I. Lmma 4 If OP T (L ) = 1, thn OP T (I) = nr E. Proof : From Rmark 1, thr ar n = 6 s disjoint lablings that satisfy all th constraints of L. Considr th ith such labling. This dfins an allocation to th ith playr as follows: w allocat th goods S v,l(v) for ach nod v, to playr i, whr l(v) is th labl of v in this ith labling. Sinc th labling satisfis all th dgs, th corrsponding sts S v,l(v) covr all th subunivrss. To s this, fix an dg = (v, w). Th labling satisfis, hnc th labls of v and w induc th sam coloring of th shard vrtics btwn v and w, and thrfor thy

7 both corrspond to th sam partition of P, say (A j, Ā j ). Thus U is covrd by th sts S v,l(v) and S w,l(w) and th utility of playr i is r E. W can find such an allocation for vry playr, sinc th lablings ar disjoint. For th No cas, considr th following simpl allocation: ach playr gts xactly on st from vry nod. Hnc ach playr i dfins a labling, which cannot satisfy mor than 2 γs fraction of th dgs. For th rst of th dgs, th 2 sts of playr i com from diffrnt partitions and hnc can covr at most 3/4 of th subunivrs. This shows that th total utility of this simpl allocation is at most 3/4 of that in th Ys cas. In fact, w will show that this is tru for any allocation. Lmma 5 If OP T (L ) 2 γs, thn OP T (I) (3/4 + ɛ)nr E, for som small constant ɛ > 0 that dpnds on s. Proof : Considr an allocation of goods to th playrs. If playr i rcivs sts S 1,..., S l, thn his utility p i can b dcomposd as p i = p i,, whr p i, = ( j S j ) U Fix an dg (u, v). Lt m i b th numbr of goods of th typ S u,j for som j. Lt m i b th numbr of goods of th typ S v,j for som j, and lt x i = m i + m i. Lt N b th st of playrs. For th dg = (u, v), lt N1 b th st of playrs whos sts covr th subunivrs U and N2 = N\N 1. Lt n 1 = N 1 and n 2 = N 2. Not that for i N 1, th contribution of th x i sts to p i, is r. For i N2, it follows that th contribution of th x i sts to p i, is at most (1 1 2 x )r by th proprtis of th partition systm of this i dg2. Hnc th total utility drivd by th playrs from th subunivrs U is p i, r + (1 1 2 x )r i i i N 1 In th YES cas, th total utility drivd from th subunivrs U was nr. Hnc th loss in th total utility drivd from U is nr r i N 1 i N 2 By th convxity of th function 2 x, w hav that r n 2 2 i N 2 (1 1 2 x i )r = r i N 2 x i n x i N2 i But not that i N1 x i 2n 1, sinc playrs in N 1 covr U and thy nd at last 2 sts to do this. Thrfor i N2 x i 2n 2 and r n 2 /4. Th total loss is r/4 n 2 2 To us th proprty of P, w nd to nsur that x i 3 s. Howvr sinc i N2, vn if x i > 3 s, th distinct sts A j or Ā j that h has rcivd through his xi goods ar all from diffrnt partitions of U and hnc thy can b at most 3 s.

8 Hnc it suffics to prov n 2 (1 ɛ)n E, or that n 1 ɛn E. For an dg (u, v), lt N, s 1 b th st of playrs from N1 typ S u,j or S v,j. Lt N,>s 1 = N1, s \N1. who hav at most s goods of th n 1 = N,>s 1 + N, s 1 2n E + s N, s 1 whr th inquality follows from th fact that for th dg w cannot hav mor than 2n/s playrs rciving mor than s goods from u and v. Claim 6, s N1 < δn E, whr δ c s2 γs, for som constant c. Proof : Suppos that th sum is δn E for som δ 1. Thn it can b asily sn that for at last δ E /2 dgs, N, s 1 δn/2. Call ths dgs good dgs. Pick a playr i at random. For vry nod, considr th st of goods assignd to playr i from this nod, and pick on at random. Assign th corrsponding labl to this nod. If playr i has not bn assignd any good from som nod, thn assign an arbitrary labl to this nod. This dfins a labling. W look at th xpctd numbr of satisfid dgs. For vry good dg = (u, v), th probability that is satisfid is at last δ/2s 2, sinc has at last δn/2 playrs that hav covrd U with at most s goods. Sinc thr ar at last δ E /2 good dgs, th xpctd numbr of satisfid dgs is at last δ 2 E /4s 2. This mans that thr xists a labling that satisfis at last δ 2 E /4s 2 dgs. But, sinc OP T (L ) 2 γs, w gt δ c s2 γs, for som constant c. W finally hav n 1 2n E s + δn E ɛn E whr ɛ is som small constant dpnding on s. Thrfor th total loss is 1 (1 ɛ)nr E 4 which implis that OP T (I) (3/4 + ɛ)nr E. Givn any ɛ > 0, w can choos s larg nough so that Lmma 5 holds. From Lmmas 4 and 5, w hav: Thorm 7 For any ɛ > 0, thr is no polynomial tim (3/4 + ɛ)-approximation algorithm for th allocation problm with monoton submodular utilitis, unlss P = NP. 4 A Hardnss of 1-1/ In this sction w obtain a strongr rsult by using a k-provr proof systm (i.., a labl covr problm on hyprgraphs) for MAX-3-COLORING. Th nw proof systm is obtaind in a similar mannr as th proof systm for MAX-3-SAT by Fig [9].

9 Lt G b an instanc of GAP-MAX-3-COLORING(c). From th graph G, w will dfin a nw labl covr instanc. Th labl covr instanc is now dfind on a hyprgraph H instad of a graph. Lt s and k b constants to b dtrmind latr. Th hyprgraph H consists of k layrs of vrtics, V 1,..., V k. To vry layr V i, w associat a binary string C i {0, 1} s of wight s/2, with th proprty that th Hamming distanc btwn any 2 strings is at last s/3. On can obtain such a collction of strings by using th codwords of an appropriat binary cod (s [9] for mor dtails). Notic that ach C i dfins a partition of th indics {1,..., s} into 2 sts A i, B i, ach of cardinality s/2, whr an indx l blongs to A i (rsp. B i ) if th l-th bit of C i is 1 (rsp. 0). W will rfr to th vrtics of H as nods. On diffrnc from Sction 3 is that now a nod of H will contain both dgs and vrtics of G. To b mor spcific, a nod v in V i is an ordrd s-tupl v = (v 1,..., v s ), whr for l {1,..., s}, if l A i, thn v l is an dg of G, othrwis it is a vrtx of G. Clarly thr ar at most n O(s) nods in ach layr V i and sinc k and s ar constants, th siz of H is polynomial in th siz of G. A labl of a nod v in V i will b a propr coloring of th s/2 dgs corrsponding to v and a coloring of th s/2 vrtics corrsponding to v. Hnc thr ar 6 s/2 3 s/2 distinct labls. For tchnical rasons w mak 6 s/2 3 s/2 copis of ach labl so that in total w hav 6 s labls in vry nod. Edgs of th hyprgraph H hav cardinality k and contain on nod from ach layr. For vry ordrd tupl of s dgs ( 1,..., s ), of G and a choic of s vrtics (u 1,..., u s ), on from ach i, w will hav a hyprdg (v 1,..., v k ) in H, with v i V i if and only if th nods v 1,..., v k satisfy th following: 1. v l i = l if l A i. 2. v l i = u l if l B i. W will call th vrtics u 1,..., u s th shard vrtics of th hyprdg (v 1,..., v k ). Givn a labling to th nods of H, lt (l(v 1 ),..., l(v k )) b th labls of th hyprdg = (v 1,..., v k ). W will say that is wakly satisfid if thr xists a pair of nods v i, v j that agr on th coloring of th shard vrtics as inducd by thir labling. W will call th pair of labls (l(v i ), l(v j )) a consistnt pair with rspct to th hyprdg and th labling. W will say that a hyprdg is strongly satisfid if for vry pair v i, v j, (l(v i ), l(v j )) is consistnt. This complts th dscription of th labl covr instanc L. Lt OP T wak (L) (rsp. OP T strong (L)) b th maximum fraction of hyprdgs that can b wakly (rsp. strongly) satisfid by any labling. Th following lmma is ssntially Lmma 5 in [9]. Lmma 8 It is NP-hard to distinguish btwn: YES Cas: OP T strong (L) = 1 NO Cas: OP T wak (L) k 2 2 γs, for som constant γ > 0. Rmark 2 In th YES Cas of Lmma 8, thr ar 6 s disjoint lablings that strongly satisfy all th hyprdgs. This follows from a similar argumnt as for Rmark 1. To dfin th instanc of th allocation problm, w will first construct a st systm as in Sction 3. For this w will nd a mor gnral notion of a partition systm:

10 Lmma 9 ([9]) Lt U = [k] n. W can construct a partition systm on U of th form P = {(A 1 1,..., A1 k ), (A2 1,..., A2 k ),..., (An 1,..., An k )}, with th proprtis that 1. For i = 1,..., n, A i j = U. 2. Any collction of l n sts, on from ach partition, covrs xactly (1 (1 1/k) l ) U lmnts. For vry hyprdg, w will hav a sparat subunivrs U. Lt n = 6 s b th numbr of labls of ach nod. For ach hyprdg w construct a partition systm P on th subunivrs U as in Lmma 9. Lt P = {(A 1,1,..., A 1,k ), (A 2,1,..., A 2,k ),..., (A n,1,..., A n,k )}. Notic that for a hyprdg = (v 1,..., v k ), w can always find n disjoint lablings of th nods v 1,..., v k that strongly satisfy th hyprdg. This follows from th fact that thr ar 6 s propr colorings of an s-tupl of dgs of G and for ach such coloring w can pick a labl from ach nod v i that rspcts this coloring. Du to th multipl copis of ach distinct labl, w in fact hav mor than n lablings that strongly satisfy. W arbitrarily pick n of ths disjoint lablings (not that any othr labling that strongly satisfis is isomorphic to on of th n lablings pickd). Assuming som arbitrary ordring among th n lablings, w associat th jth partition of P with th jth labling of, for vry. If (l j 1,..., lj k ) is th jth labling of and (A j,1,..., A j,k ) is th jth partition of P w will also say that th st A j,i corrsponds to th labl l j i of v i. W can now dfin our st systm. W will hav on st S v,i for vry nod v and labl i. Lt v V l for som l [k]. For an dg that contains nod v, suppos labl i is in th jth labling of. W will thn includ th st A j,l from th jth partition in S v,i. Hnc S v,i is th following union of sts: S v,i = :v whr j (i) is th labling of dg that contains i. A j (i),l As in Sction 3, th instanc of th allocation problm contains n = 6 s playrs with th sam submodular utility function. Th goods ar th sts S v,i and th utility of a playr for a collction of sts is th cardinality of thir union. Lt I dnot th instanc of th allocation problm and lt OP T (I) b th optimal solution of I. Lt r = U and lt E b th st of th hyprdgs of H. Our hardnss rsult is stablishd with th following lmmas, which w prov in th Appndix. Lmma 10 If OP T strong (L) = 1, thn OP T (I) = nr E. Lmma 11 If OP T wak (L) k 2 2 γs, thn OP T (I) (1 1/ + ɛ)nr E, whr ɛ > 0 is som small constant dpnding on s and k. Givn any ɛ > 0, w can choos larg nough constants s, k so that Lmma 11 holds. Hnc w gt: Thorm 12 For any ɛ > 0, thr is no polynomial tim (1 1 + ɛ)-approximation algorithm for th allocation problm with monoton submodular utilitis, unlss P=NP.

11 5 Conclusion and Opn Problms In this papr, w hav provd a (1 1/ 0.632)-hardnss of approximation in th valu qury modl. Thr is a gap btwn th uppr and lowr bounds in both th valu qury and dmand qury modl. It would b intrsting to narrow ths gaps. It will also b intrsting to obtain truthful mchanisms with good approximation guarants. Acknowldgmnts W would lik to thank Amin Sabri and Vahab Mirrokni for usful discussions and Shahar Dobzinski for commnts. Rfrncs [1] N. Andlman and Y. Mansour. Auctions with budgt constraints. In SWAT, [2] A. Archr, C. Papadimitriou, K. Talwar, and E. Tardos. An approximat truthful mchanism for combinatorial auctions with singl paramtr agnts. In SODA, pags , [3] S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szgdy. Proof vrification and hardnss of approximation problms. In FOCS, pags 14 23, [4] Y. Bartal, R. Gonn, and N. Nisan. Incntiv compatibl multi unit combinatorial auctions. In TARK, pags 72 87, [5] L. Blumrosn and N. Nisan. On th computational powr of ascnding auctions 1: Dmand quris. In ACM Confrnc on Elctronic Commrc, [6] P. Cramton, Y. Shoham, and R. Stinbrg(ditors). Combinatorial Auctions. MIT Prss, Forthcoming (2005). [7] S. Dobzinski, N. Nisan, and M. Schapira. Approximation algorithms for combinatorial auctions with complmnt-fr biddrs. In STOC, [8] S. Dobzinski and M. Schapira. An improvd approximation algorithm for combinatorial auctions with submodular biddrs. Working papr, [9] U. Fig. A thrshold of lnn for approximating st covr. Journal of th ACM, 45(4): , [10] F. Gul and E. Stacchtti. Walrasian quilibrium with gross substituts. Journal of Economic Thory, 87:66 95, [11] R. Holzman, N. Kfir-Dahav, D. Mondrr, and M. Tnnnholtz. Bundling quilibrium in combinatorial auctions. Gams and Economic Bhavior, 47: , [12] B. Lhmann, D. Lhmann, and N. Nisan. Combinatorial auctions with dcrasing marginal utilitis. In ACM Confrnc on Elctronic Commrc, 2001.

12 [13] D. Lhmann, L. O Callaghan, and Y. Shoham. Truth rvlation in approximatly fficint combinatorial auctions. In ACM Confrnc on Elctronic Commrc, [14] C. Lund and M. Yannakakis. On th hardnss of approximating minimization problms. Journal of th ACM, 41(5): , [15] N. Nisan and I. Sgal. Th communication rquirmnts of fficint allocations and supporting lindahl prics. To appar in Journal of Economic Thory, prliminary vrsion availabl at noam/mkts.html, [16] C. Papadimitriou and Yannakakis. Optimization, approximation and complxity classs. Journal of Computr and Systm Scincs, 43: , [17] R. Raz. A paralll rptition thorm. SIAM Journal of Computing, 27(3): , [18] T. Sandholm. An algorithm for optimal winnr dtrmination in combinatorial auctions. In IJCAI, Appndix Proof of Lmma 10 : Sinc OP T strong (L) = 1, considr a labling that strongly satisfis all th hyprdgs. By th discussion abov, w can always pick a labling such that whn rstrictd to th nods of an dg, it corrsponds to on of th n disjoint lablings of that dg. Lt l(v) b th labl of ach nod. Pick a playr and allocat to him all th sts {S v,l(v) }. W claim that th sts covr th subunivrs U for vry dg and th utility of th playr is thrfor r E. To s this, fix an dg = (v 1,..., v k ). Sinc th labling strongly satifis th dg, it corrsponds to som partition of th partition systm P, say th jth partition. Hnc for i = 1,..., k, th st A j,i which corrsponds to labl l(v i) is containd in S vi,l(v i ). Thus th playr covrs th ntir subunivrs U with th sts S vi,l(v i ). Sinc this is tru for vry dg, his utility is xactly r E. By Rmark 2 w can rpat th abov for all th 6 s playrs. Proof of Lmma 11 : Considr an allocation of th goods to th playrs, i.., an allocation of th labls of ach nod. W dcompos th utility p i of playr i as: p i = p i,, whr p i, is as in Sction 3. For a nod v and a playr i, lt m v i b th numbr of sts of th typ S v,j that playr i has rcivd. Fix an dg = (v 1,..., v k ). Lt x i = k l=1 mv l i. Dfin th st of playrs: N 1 = {i : v j, v l such that i has a pair of consistnt labls for ths 2 nods} Lt N2 = N\N 1, and lt n 1 = N 1, n 2 = N 2. Trivially, for i N 1, th contribution of th x i sts to p i, is at most r. For i N2, th x i sts of th typ S v l,j do not contain vn on pair of labls which ar consistnt for som pair of nods in. For ach st S vl,j that playr i has rcivd, lt A t,l b th st from th partition systm P containd in S vl,j. It follows that th sts A t,l corrsponding to th labls of playr i com from diffrnt partitions of U. Thrfor, by Lmma 9, w gt that th sts S vl,j covr xactly 1 (1 1 k )x i fraction of th subunivrs U. Hnc th total utility drivd by th playrs from th subunivrs U is p i, r + ( ) 1 (1 1 k )x i r i i N 1 i N 2

13 Th loss in th total utility compard to th YES cas is: nr r (1 (1 1 k )x i )r = r (1 1 k )x i i N 1 i N 2 i N 2 By th convxity of th function (1 1 k )x, w hav that rn 2(1 1 k ) i N 2 x i n 2 (1) Lt N, k2 1 b th st of playrs from N1 who hav at most k2 goods of th typ S vl,j. Lt N,>k2 1 = N1, k2 \N1. n 1 = N,>k2 1 + N, k2 1 kn E k 2 + N, k2 1 whr th inquality follows from th fact that for th dg w cannot hav mor than n/k playrs rciving mor than k 2 goods from th nods v 1, v 2,..., v k. Claim 13, k2 N1 < δn E, for δ ck 3 2 γs, for som constant c. Proof : Th proof is similar to that of Claim 6. If w assum th contrary to th statmnt thn w can find a labling which wakly satisfis mor than k 2 2 γs fraction of th dgs, a contradiction. Hnc n 1 n E k +δn E, which implis n 2 (1 β)n E, for som small constant β > 0. In Sction 3, this sufficd to obtain th hardnss rsult of 3/4, bcaus i N2 x i 2n 2. Hr a similar argumnt would nd that i N2 x i kn 2, which may not b tru for vry dg bcaus playrs in N1 ar only wakly satisfying. Howvr, w will s that for most dgs, is still small. i N 2 x i Sinc n 1 βn E, it follows that for at last a 1 β fraction of th dgs, n 2 (1 β)n. Call ths dgs good. For ach good dg : i N 2 x i n 2 kn (1 β)n k(1 + β ) for som small constant β > 0. From (1), w gt that for vry good dg th loss rn 2 (1 1 k )k(1+β ) rn 2 (1 β ) 1, for som small constant β > 0. Summing th loss ovr all th good dgs, w gt that th total loss in utility is at last r :is good (1 β)n(1 β ) 1 n r E (1 β) 2 (1 β ) 1 nr E (1 ɛ) whr ɛ > 0 is som small constant. Hnc th total utility is at most (1 1 + ɛ)nr E