Frugal and Truthful Auctions for Vertex Covers, Flows, and Cuts

F rugal and T ruthful Auctions for V ertex Co v ers, Flo ws, and Cuts ∗ Da vi d Kemp e Department of Comput er Science, Universit y of Southern C alifo rnia , CA 9008 9-078 1, USA dkem pe@u sc.e du Mah yar Salek Department of Comput er Sci en ce, Universit y o f Souther n Cali forni a, CA 9 0 089-0 781, USA sale k@us c.ed u Cristo pher Mo or e Comput er Science Departm en t and Department of Physics and Astr o nomy , Universit y o f New Mexico , Albuquer que, NM 87 131-0 001, USA and Sa n ta F e Inst itute, Santa F e NM 8750 1, USA moor e@cs .unm .edu Octob er 31, 2018 Abstract W e study truthful m echanisms for hiring a team of agen ts in three classes of set systems: V ertex Co v er auctions, k -ﬂo w auctions, a n d cut auctions. F or V ertex Cov er auctions, the ve r tices are o wned by selﬁsh and r ational agen ts, and the auctioneer w ants to p urc h ase a verte x co v er from them. F or k -ﬂ o w a u ctions, th e edges are o w ned b y the agen ts, and the auctioneer wan ts to p urc h ase k edge-disjoin t s - t paths, f or giv en s an d t . In the same setting, for cu t auctions, the auctioneer w an ts to purc h ase an s - t cut. On ly th e agen ts kno w their costs, and the auctioneer n eeds to select a feasible set and pa ym en ts based on bids made by th e agen ts. W e present constan t-comp etitiv e truthf ul m ec hanisms for all th ree set systems. That is, the maxim um o ve r p a yment of the mec hanism is within a constan t factor of the maxim um o ve r p a yment of an y truthfu l mec han ism , for every set sys tem in the ∗ A preliminary version o f this article app eared in the Pro ceeding s of FOCS 201 0 [19]. 1 class. The mec h anism for V ertex Co v er is based on scaling eac h bid b y a multiplier deriv ed fr om the dominan t eigen vec tor of a certain matrix. Th e mechanism for k -ﬂows prun es the graph to b e minimally ( k + 1) -connected, and then applies the V ertex Cov er mec hanism. Similarly , the mec hanism for cuts con tracts the graph unti l all s - t paths ha ve length exactly 2, and then ap p lies the V ertex Co ver mec hanism. 1 In tro ducti on Man y tasks require the joint allo cation o f m ultiple resources b elonging to diﬀeren t bidders. F or instance, consider the ta sk of routing a pac k et through a net w o r k whose edges are o wned b y diﬀeren t ag en ts. In this setting, it is necessary to obta in usage righ ts for m ultiple edges sim ultaneously from the agen ts. Similarly , if the agen ts o wn t he v ertices of a graph, and w e w ant to monitor all edges, w e need the righ t to install monitoring devices on no des, and again obtain these righ ts from distinct agen ts. Pro viding access to edges or no des in suc h settings mak es the agents incur a cost c e , whic h the agen ts should b e paid for . A conv enien t w a y to determine “a ppropriate” prices to pay the agen ts is b y wa y of auctions , wherein the agen ts e submit bids b e to an a uctione er , who selects a fe asib le subset S of a gen ts to use, and determines prices p e to pa y the agen ts. The most basic case is a single-item auction. The auctioneer requires the service of a n y one of the a gen ts, and their services are in terc hangeable. Single- it em auctions hav e a long history of study , and are fairly w ell understo o d [20, 21]. Motiv ated b y applications in computer net w orks and electronic commerce, sev eral recen t pap ers ha ve considered the extension to a setup termed hiring a te am of agen ts [3 , 11, 12, 18, 28]. In this setting, there is a collection of fe asible sets , eac h consisting of one or more agent. The auctioneer, based on the agents’ bids b e , selec ts one feasible set S , and pa ys eac h a g en t e ∈ S a price p e . Some of the w ell-studied special cases of set systems are p ath auctions [3, 12, 18, 24, 30], in whic h the feasible sets are pa ths from a given source s to a giv en sink t , and sp anning tr e e auctions [4, 14, 18, 28], in whic h the feasible sets are spanning trees of a connected graph. In b oth cases, the agen ts are the edges o f the graph. In this pa p er, we extend the study to more complex examples of set systems, namely: 1. V ertex Co vers : The a g en ts are the vertic es of the graph G , and the auctioneer needs to select a v ertex co v er [5, 11, 28]. Not only are v ertex cov ers o f in terest in their o wn righ t, but they giv e a k ey primitiv e for man y other set systems as well, an approac h w e explore in depth in this pap er. 2. Flo ws: The agen ts are the edges of G , and the auctioneer w an ts to select k edge-disjoin t paths from s to t . Th us, this scenario g eneralizes path auctions; the generalization turns out to require signiﬁcan t new techniq ues in the design and analysis of mec hanisms. 3. Cuts: In the same setting a s for ﬂo ws, the auctioneer w ants to purc hase an s - t cut. In c ho osing a n a uction mec hanism fo r a set system, the auctioneer needs to tak e in to accoun t tha t the agen ts are selﬁsh. Ideally , the auctioneer w ould like t o kno w the a g en ts’ 2 true costs c e . How ev er, the costs are priv ate information, and a r a tional and selﬁsh agent will submit a bid b e 6 = c e if doing so leads to a higher proﬁt. The ar ea of me chanism desi g n [23, 24, 25] studies the design of auctions for selﬁsh and rational agen ts. W e are interes ted in designing trut hful (or inc entive - c omp atible ) a uction mec hanisms: auctions under whic h it is alw ays optimal f or selﬁsh agen ts to rev eal their priv ate costs c e to the auctioneer. Suc h mec hanisms are so cietally desirable, b ecause they mak e the computation of strategies a trivial task for the ag en ts, and ob viate t he need for gathering information ab out t he costs or strategies of comp etitors. They are also desirable from the p oin t of view of analysis, as t hey allo w us to iden tify bids with costs, and let us disp ense with an y kinds of assumptions ab out t he distribution of a gen ts’ costs. Th us, the outcomes of truthful mec hanisms are stable in a stronger sense than Nash equilibria, and ma y giv e bidders more conﬁdence that the righ t outcome will b e reac hed. F or this reason, truthful mec hanism design has b een a mainsta y of game theory for a long time. It is well kno wn that any truthful mec hanism will ha ve to pa y agents more than their costs at times; in this pap er, w e study mec hanisms approximately minimizing the “ov erpay ment.” The ratio b etw een the paym ents of the “ b est” truthful mech a nism and natural lo wer b ounds has b een termed the “Price of T ruth” b y T alw ar [28], a nd studied in a n um b er of recen t pap ers [3, 4, 11, 12, 14, 18, 28, 30]. In particular, [18] a nd [11] deﬁne and analyze diﬀeren t natural measures of lo we r b ounds on pay ments , a nd deﬁne the notions of frugalit y ratio and comp etitiv eness. The frugality r atio of a mec ha nism is the w orst- case ratio of pa ymen ts to a natural low er b ound (formally deﬁned in Section 2), maximized ov er all cost v ectors of the agen ts. A mec hanism is c omp etitive for a class of set syste ms if its frugality ratio is within a constan t factor of the f r ug alit y ratio of the b est truthful mec hanism, f or a l l set systems in the class. 1.1 Our Cont ribu tions In this pa p er, w e presen t no v el f r ug al mec hanisms for three general classes of set systems: V ertex Co v ers, k -Flows , and Cuts. V ertex Cov er auctions can b e considered a v ery na t ural primitiv e for more complicated set systems . Under the na t ural assumption that there are no isolated v ertices, they capture set systems with “minimal comp etitio n”: if the a uction mec hanism decides to exclude an agen t v fro m the selected set, this immediately forces the mec hanism to include all of v ’s neigh b ors, thus g iving these neigh b ors a monop o ly . Th us, a diﬀeren t in terpretation of V ertex Co v er auctions is that they capture any set system whose feasible sets can b e c haracterized b y p ositiv e 2SA T form ulas: eac h edge ( i, j ) corresp onds to a clause ( x i ∨ x j ), stating that an y feasible set mus t include at least one of agen ts i and j . Our mec hanism for V ertex Co v er works a s follow s: based solely on the structure of the graph G , w e deﬁne an appropriate matr ix K and compute its dominan t eigen v ector q . After agen ts submit their bids b v , the mechanis m ﬁrst scales eac h bid to c ′ v = b v /q v , and then simply runs the V CG mec hanism [29, 9, 15] with these mo diﬁed bids. W e prov e that this mec hanism has a fr ug alit y rat io equal t o the largest eigen v alue α o f K , and t hat t his is within a factor o f 2 of the frugalit y ratio of an y mec hanism. The low er b o und is based on pairwise comp etition b etw een adjacen t bidders for any truthful mec hanism, and in a sense 3 can b e considered the natural culmination of the lo w er b ound tec hniques of [12, 18]. The upp er b ound is based on carefully balancing a ll p ossible worst cases of a single non-zero cost against eac h other, and sho wing that the w orst case is indeed one of these cost v ectors. W e stress here that the mec hanism do es not in general run in p olynomial time: the en tries of K are deriv ed from fra ctional clique sizes in G , whic h are kno wn to b e hard to compute, ev en appro ximately . W e discuss the issue of p o lynomial time brieﬂy in Section 6. Based on o ur V ertex Co ver mec hanism, w e presen t a general metho dolo g y for designing frugal truthful mec hanisms. The idea is to tak e the orig inal set system , and prune agents from it un til it has “minimal comp etition” in the ab ov e sense; subsequen tly , t he V ertex Co v er auction can b e in v oked. So long as the pruning is “comp osable” in the sense of [1] (see Section 3), the resulting auction is truthful. The crux is then to pro v e that the pruning step (whic h remov es a signiﬁcant amoun t of comp etition) do es not increase the low er b ound on pa ymen ts to o m uch. W e illustrate the p o wer of this a pproac h with t w o examples. 1. F or the k -ﬂow problem, w e sho w that pruning the gr aph to a minim um- cost ( k + 1) s - t -connected graph H is comp osable, a nd increases the lo wer b ound at most by a factor o f k + 1. Hence, we obtain a 2( k + 1 )-comp etitiv e mec hanism. Establishing the b ound of k + 1 requires signiﬁcan t tec hnical eﬀort. 2. F or the cut pro blem, w e show that pruning the graph to a minim um- cost set of edges suc h that eac h s - t pa t h is cut at least t wice giv es a comp o sable selection rule. F urt her- more, it increases the low er b ound by at most a factor of 2, leading to a 4 -comp etitiv e mec hanism. F or the pruning step, w e deve lo p a primal-dual algorithm generalizing the F ord- F ulke rson Minimum-Cut algorithm. W e note that while the V ertex Co v er mec hanism is in general not p olynomial, f o r b oth sp ecial cases deriv ed here, the running time is in fact p olynomial. 1.2 Relationship to P ast and P arallel W ork As discussed ab o ve , a line of recent pap ers [3, 4, 11, 12, 14, 18, 28, 30] analyze frugality of auctions in t he “hiring a team” setting, where the auctioneer w ants to obtain a feasible set of agen ts, while paying not muc h more than necessary . In t his context, the pap ers by Karlin, Kemp e, and T amir [18] and Elkind, Goldb erg, and Goldb erg [11] are particularly related to our w ork. Karlin et al. [18] in tro duce the deﬁnitions of fruga lity and comp etitiv eness whic h we use here. They also give comp etitiv e mec hanisms for path auctions, and for so-called r -out- of- k systems , in whic h the a uctioneer can select an y r out of k disjoin t sets of agen ts. A t the heart of b oth mec hanisms is a mec hanism for r - out-of-( r + 1) systems. Our mech a nism for V ertex Co v ers can b e considered a natural generalization of this mec hanism. F urthermore, b oth r -out- of- k system s and pat h auctions are sp ecial cases o f r -ﬂow s, since c ho osing an r -ﬂo w in a graph consisting of k vertex - disjoin t s - t paths is equiv alen t to an r -out-of - k system. Our approac h of pruning the graph is similar in spirit to the approach in [18], where gra phs were 4 also ﬁrst pruned to b e minimally 2- connected, and set systems we re reduced to r -o ut-of- ( r + 1) systems. How ev er, the combinatorial structure o f k -ﬂow s mak es t his pruning ( and its analysis) m uc h more in volv ed in our case. Elkind et al. [11] study truthful mec hanisms for V ertex Cov er. They presen t a p olynomial- time mec hanism with frugality ratio b ounded b y 2∆, where ∆ is the maximum degree of the gra ph, and also sho w that there exist graphs where the b est truthful mec hanism m ust ha v e frugality ratio at least ∆ / 2. Notice, ho wev er, that this do es not guar a n tee that the mec hanism is comp etitiv e. Indeed, there are graphs where the b est truthful mec hanism has frugality ratio signiﬁcan tly smaller than ∆ / 2, a nd our goal is to hav e a mec hanism whic h is within a constan t factor of b est p ossible for every g r aph . Sev eral recen t pap ers hav e extended the pr o blem of hiring a team of agen ts in v arious directions. Cary , Flaxman, Hartline, and K arlin [6] com bine truthful auctions for hiring a team with rev en ue-maximizing auctions fo r selling items. Du, Sami, and Shi [1 0] and Iw asaki, Kemp e, Saito, Salek, and Y ok o o [1 7] study pat h auctions under the a dditio nal requiremen t that not only should they b e truthful, but false-name pr o of : a gen ts o wning m ultiple edges ha v e no incen tive to claim that these edges b elong to diﬀeren t agen ts. Du et al. sho w that there are no false-name pro of mec ha nisms that a re also P areto- eﬃcien t, and Iwas a ki et al. analyze the frugality ra tio of false-name pro of mec hanisms, show ing exp onential lo wer b ounds. Results very similar to ours hav e b een deriv ed indep enden tly and sim ulta neously by Chen, Elkind, Grav in, a nd P etrov [7]. Both pap ers ﬁrst deriv e mec hanisms for V ertex Co ver auctions. Our mec hanism is based on scaling the agen ts’ bids b y the en tries of the dominant eigen v ector of a scaled adjacency matrix. It has constan t comp etitiv e ratio for all g raphs, but may no t run in p olynomial time. The mec hanism of Chen et al., on the other ha nd, uses eigen v ectors o f the unscaled adjacency matrix. It may not b e constan t comp etitiv e on some inputs, but it alw ay s runs in p olynomial time. Chen et al. also prop ose the approa c h of reducing other set systems to V ertex Cov er instances, called “ Pruning-Lifting Mec hanisms” there. In particular, they deriv e the same mec hanism as the presen t pap er for k -ﬂows , with similar k ey lemmas in the pro of. While their V ertex Co ver mec hanism is diﬀeren t fro m ours in general, on inputs deriv ed from ﬂo w and cut problems, the scaling facto r in our matrix is the same for all en tries, and the mec hanisms therefore coincide. In particular, the mec hanisms in b oth pap ers are t h us comp etitiv e and run in p olynomial time. While the mec hanism of Chen et al. [7] may not alw ay s b e comp etitive due to the lac k of scaling f actors in the matrix, their pro of of a low er b ound in v olv es a clev er application of Y o ung’s Inequalit y , a nd th us a v o ids losing t he factor of 2 in our lo w er b o und. Th us, whenev er their mec hanism coincides with ours, b oth mec hanisms ar e optimal. In particular, this also implies that the k -ﬂow mec hanism of the presen t pap er is ( k + 1)-comp etitiv e and our mec hanism for s - t cuts is 2-comp etitive . Moreov er, they pro v e stronger b ounds on the k -ﬂow mec hanism: when compared a g ainst the low er b o und from [1 1] (used in this pap er, and deﬁned formally in Section 2), the mec hanism is in fact optimal. Finally , in colla b oration with the authors of [7], w e recen tly show ed that Y oung ’s In- 5 equalit y can b e applied to the analysis of our V ertex Cov er mec hanism, removing the factor of 2 from the low er b ound. In other w ords, w e sho w that our V ertex Cov er mec hanism is indeed optimal for all V ertex Co v er instances. This result will be included in a joint full v ersion of b ot h pap ers. 2 Preliminaries A set system ( E , F ) has n ag e nts (or elements ), and a collection F ⊆ 2 E of f e asible sets . W e call a set system m o nop o ly-fr e e if no elemen t is in a ll feasible sets, i.e., if T S ∈F S = ∅ . The three classes of set syste ms studied in this pap er are: 1. V ertex Cov ers: here, the agen ts are the vertic es of a graph G , and F is t he collection of all vertex co v ers of G . T o av oid confusion, we will denote the agen ts b y u , v instead of e in this case. Notice that every V ertex Co ve r set system is monop oly-f r ee. 2. k -ﬂows : here, w e are giv en a graph G with source s and sink t . The agen ts are the e dge s of G . A set of edges is feasible if it con tains at least k edge-disjoin t s - t paths. A k -ﬂo w set system is monop o ly-free if and only if the minim um s - t cut cuts at least k + 1 edges. 3. Cuts: With the same setup as for k -ﬂo ws, a set o f edges is feasible if it con tains an s - t cut. Th us, the set system is monop oly-free if and only if G con tains no edge fro m s to t . The set system ( E , F ) is common knowled g e to the a uctioneer and all agen ts. Each agen t e ∈ E ha s a c ost c e , which is priv ate, i.e., known only to e . W e write c ( S ) = P e ∈ S c e for the total cost of a set S of agen t s, and also extend this notation to other quan tities (suc h as bids or pa ymen ts). A me chanism for a set system pro ceeds as follow s: 1. Eac h agen t submits a sealed bid b e . 2. Based on the bids b e , the auctioneer sele cts a feasible set S ∈ F a s the winner, and computes a paym ent p e ≥ b e for each agen t e ∈ S . The agents e ∈ S are said to win , while all other agen ts lose . Eac h agent, kno wing the algo rithm for computing the winning set and the pa ymen t s, will c ho ose a bid b e maximizing her o wn pr oﬁt , whic h is p e − c e if t he agent wins, a nd 0 otherwise. W e are intere sted in mec hanisms where self-in terested agents will bid b e = c e . More precisely , a mec hanism is truthful if , for any ﬁxed v ector b − e of bids b y all other agents , e maximizes her proﬁt b y bidding b e = c e . If a mec hanism is kno wn t o b e truthful, w e can use b e and c e in terc hangeably . It is w ell-know n [3, 21] that a mec hanism is trut hf ul only if its selection rule is mon otone in the follo wing sens e: if all other agen ts’ bids sta y t he same, then a losing agen t cannot b ecome a winner b y raising her bid. Once the selection rule is ﬁxed, there is a unique payme nt sc heme to make the mec hanism truthful. Namely , eac h agen t is paid her thr eshold b id : the suprem um of all winning bids she could hav e made giv en the bids of all other agen ts. 6 2.1 Nash Equilibria and F ru galit y Ratios T o measure ho w m uch a truthful mec hanism “ov erpays ,” we need a natur a l b ound t o compare the pa ymen ts to. Karlin et al. [18] prop osed using as a b ound the solutio n of a natura l minimization problem. Let S b e the cheapest feasible set with resp ect to the true costs c e ; ties are brok en lexicographically . Minimize ν − ( c ) := P e ∈ S x e sub ject to x e ≥ c e for all e ∈ S x e = c e for all e / ∈ S P e ∈ S x e ≤ P e ∈ T x e for all T ∈ F F or ev ery e ∈ S , there is a T e ∈ F , e / ∈ T e suc h that P e ′ ∈ S x e ′ = P e ′ ∈ T e x e ′ (1) The intuition for this optimization problem is that it captures the bids o f agen ts in the c heap est “Nash Equilibrium” of a ﬁrst-price auction with full information, under the assumption that the actual cheapest set S wins, and the lo sing ag en ts all bid their costs. That is, the mec hanism selects t he che ap est set with resp ect to the bids x e , and pa ys each winning agen t her bid x e . The ﬁrst constrain t captures individual r a tionalit y . The thir d constrain t states t ha t the bids x e are suc h that S still wins, and the ﬁna l constrain t states that f or eac h winning a gen t, there is a tight set prev enting her fro m bidding higher. That is, if e increases her bid, the buy er will select a set T excluding e instead of S . W e say that a v ector x is fe asible if it satisﬁes all these constrain ts. While this optimization problem is inspired b y the analogy of Nash Equilibria, it should b e noted that ﬁrst-price auctions do not in general hav e Nash Equilibria due to tie-br eaking issues (see a more detailed discussion in [16, 18]). Elkind et a l. [11] and Chen and Karlin [8] observ ed that the quan tity ν − ( c ) has sev eral undesirable non-monotonicit y prop erties. F or instance, adding new feasible sets to the set system, and th us increasing the a moun t of comp etition b et w een agents, can sometimes lead to higher v a lues of ν − ( c ). Similarly , lo we ring t he costs of losing agen ts, o r increasing the costs of winning a g en ts, can sometimes increase ν − ( c ). F urthermore, ν − ( c ) is NP-hard to compute ev en if the set system is the set of all s - t paths [8]. Instead, Elkind et al. [11] prop ose replacing the minimization b y a maximization in the ab ov e optimizatio n problem. An imp ortant adv an tage of this optimization problem is that the maximization ob j ective ensures that for ev ery e ∈ S , there is a tight set T . Th us, the maximization ob j ectiv e remov es the need for the ﬁnal constrain t, a nd turns the optimizatio n problem in to an instance of Linear Programming, whic h can b e solv ed in many cases. W e t h us obtain the follo wing deﬁnition (whic h [11] refers to as NTU max ). In tuitively , this deﬁnition captures the bids in the most exp en sive Nash Equilibrium of a ﬁrst-price auction, with t he same ca v eat as b efore ab o ut the no n-existence of equilibria. 7 Maximize ν ( c ) := P e ∈ S x e sub ject to (i) x e ≥ c e for all e (ii) x e = c e for all e / ∈ S (iii) P e ∈ S x e ≤ P e ∈ T x e for all T ∈ F (2) As stated ab o ve, a consequence of this maximization is that, for ev ery e in the winning set, there is a tigh t set T excluding e that prev en t s e from bidding higher: ∀ e ∈ S : ∃ T ∈ F : e / ∈ T and X e ′ ∈ S x e ′ = X e ′ ∈ T x e ′ . (3) W e will refer to the b ounds ν − ( c ) and ν ( c ) as buyer-optimal and buyer-p ess i mal , respec- tiv ely , throughout the pap er. Moreov er, due to the adv an ta ges discussed ab ov e, w e will use the quan tity ν ( c ) as a natur a l lo we r b o und f o r this pap er. Despite the preceding discussion, in order to emphasize the in tuitio n b ehind t he b ounds, w e will refer to the x e v alues of the LP (2) as the Nash Equilibrium bids of agen t s e , o r simply the Nash bids of e . Notice that ν ( c ) is deﬁned for all mo no p oly-free set systems . W e no w formally deﬁne the frugality ratio o f a mec hanism M fo r a set system ( E , F ), and the notion of a comp etitive mec hanism. Deﬁnition 2.1 (F rugalit y Ratio, Compet itiv e Mec hanism) L et M b e a truthful me ch- anism for the set system ( E , F ) , and let P M ( c ) deno te the total p ayments of M when the ve ctor of actual c o s ts is c . 1. The frug alit y ratio of M is φ M = sup c P M ( c ) ν ( c ) . 2. The frugality r atio of the set s ystem ( E , F ) is Φ ( E , F ) = inf M φ M , wher e the inﬁmum is taken o v er al l truthful m e chani s m s M for ( E , F ) . 3. A me chanism M is κ -comp etitiv e for a class of set s ystems { ( E 1 , F 1 ) , ( E 2 , F 2 ) , . . . } if φ M is within a factor κ of Φ ( E i , F i ) for al l i . Remark 2.2 The f rugalit y ratio o f a mec hanism is deﬁned as instance-based. The frugality ratio of a set system captures the inheren t structural complexit y of that instance, whic h can b e “exploited” with careful w o rst-case c hoices of costs. Comp etitiv eness, on the other hand, is deﬁned ov er a class of set systems. If a single mec hanism, suc h as the ones deﬁned in this pap er, is competitive, it do es as well o n eac h set system in the class as the b est mec hanism, whic h could p ossibly b e tailored to this 8 sp eciﬁc instance. The nomenclature “comp etitive ” is motiv ated b y the analo gy with online algorithms. The instance-based deﬁnition [18, 11] a llows us a more ﬁne-gr a ined distinction b et wee n mec hanisms than earlier w ork (e.g., [3, 24]), where a low er b ound in terms of a w orst case o ve r all instances w as used. As discussed a b ov e, the motiv ation for the LPs (1) and (2) was that they prov ide “natura l lo we r b o unds” on the paymen ts of any truthful mec hanism. Ho we ve r , t o the b est o f our kno wledge, it was previously unknow n whether the solutions do in fact pro vide lo wer b ounds. Indeed, it is easy to deﬁne mec ha nisms that ac hiev e arbitrarily lo we r pa ymen ts for particular cost v ectors, alb eit at the cost of signiﬁcantly higher pa ymen ts on other cost vec to r s. Here, w e establish that the ob jectiv e v alue of the LP (2) indeed do es giv e a low er b o und in t erms of the frugality ratio. This resolv es an op en question from the preliminary v ersion of this pap er [19]. Prop osition 2.3 L et ( E , F ) b e an arbitr ary set system and M a truthful and i n dividual ly r ational me chanism on ( E , F ) . T hen, φ M ≥ 1 . Pro of. Let c b e an arbitrary cost v ector. Let S ∈ F b e the set minimizing c ( S ), and x the solution to the LP (2). Let S ′ ∈ F b e the winning set for M with cost v ector x . Because M is truthful and individually rational, its pay ment P M ( x ) is at least P e ∈ S ′ x e . By the third constrain t of the LP (2), P e ∈ S ′ x e ≥ P e ∈ S x e . Finally , b y construction, we hav e that ν ( c ) = ν ( x ). T a k en tog ether, this implies that P M ( x ) ≥ X e ∈ S ′ x e ≥ X e ∈ S x e = ν ( x ) . By deﬁnition of the frugality ratio, this implies that φ M ≥ 1. 3 V ertex Cov er Au c t ions In this section, w e describ e and analyze a constan t-comp etitiv e mec hanism for V ertex Co ve r auctions. W e then show how to use it as the basis for a metho dology for designing frugal mec hanisms for other set systems . The graph is denoted by G = ( V , E ), with n ve rtices. W e write u ∼ v to denote that ( u , v ) ∈ E . Our mec hanmism is based on certain modiﬁcatio ns to the well-kno wn Vickrey - Clark e- Gro ves (V CG) mec hanism [2 9, 9, 15]. Recall that V CG alw ays selects the che ap est f easible set S with resp ect to the submitted bids b e , and pa ys eac h a g en t her threshold bid. 3.1 W eigh ting the bids with an eigen v ector The imp ortan t change to VC G in our mec hanism is that eac h agen t’s bid is scaled b y an agent-speciﬁc m ultiplier. The m ultipliers capture “ho w impor t a n t” an agent is f or the solution, roughly in the sense of how man y other agen ts can b e omitted b y including this 9 agen t. They ar e computed as en tries of the dominan t eigen v ector of a certain matrix K . As w e will see, the computation of K is NP-hard itself, so the mec hanism will in general not run in p olynomial time unless P=NP . As a ﬁrst step, our mech a nism remo v es all isolated v ertices. W e assume that the resulting graph G is connected. Let 1 v (for an y ve rtex v ) b e the v ector with 1 in co ordinate v and 0 in a ll other co ordinates. W e deﬁne ν v = ν ( 1 v ) ≥ 1 to b e the total “Nash Equilibrium ” pa ymen t of the ﬁrst-price auction in the sense of the L P (2) if a gen t v has cost 1 and all other agen ts hav e cost 0. No tice that in this case, v loses. W e prov e in Section 3 .1 that ν v is exactly the fractional clique n um b er of the graph induced by the neigh b or s of v, without v itself. This implies that unless ZPP=NP , ν v cannot b e appro ximated to within a f a ctor O ( n 1 − ǫ ) in p o lynomial time, for an y ǫ > 0. Our ina bility to compute ν v is the c hief obstacle to a constan t-comp etitive p olynomial-time mec hanism. Let A b e the adjacency matrix of G (with diag onal 0). Deﬁne D = diag(1 /ν 1 , 1 /ν 2 , . . . , 1 /ν n ), and K = D A . That is, K u,v = ( 1 /ν u if u ∼ v 0 if u 6∼ v . If w e deﬁne K ′ = D − 1 / 2 K D 1 / 2 = D 1 / 2 AD 1 / 2 , then K and K ′ ha v e the same eigen v alues, and the eigen v ectors of K a re of the form D 1 / 2 · e , where e is an eigen v ector of K ′ . Moreov er, K ′ u,v = ( 1 / √ ν u ν v if u ∼ v 0 if u 6∼ v , so K ′ is symme tric and has non-negativ e en tries. By the Perron-F rob enius Theorem, t he eigen v alues of K ’ and K are real. Since w e assume d G t o b e connected, the dominant eigen v ector of K ′ is unique and has p ositiv e en t r ies, and the same holds for K . Let α b e the largest eigen v alue of K , and q the corresp onding eigenv ector. Notice that giv en K as input, α a nd q can b e computed eﬃcien tly and without kno wledge of t he agen t s’ bids or costs. The mec hanism E V (which stands for “Eigen ve ctor Mec hanism”) is no w as follo ws: after all no des v submit t heir bids b v , the alg o rithm sets c ′ v = b v /q v , a nd computes a minim um cost v ertex cov er S with resp ect to the costs c ′ v (ties broken lexicographically). S is chosen as the winning set, and each a gen t in S is paid her threshold bid. No tice that the second step of the mec hanism again requires the solution to an NP-ha r d pro blem. E V is truthful since the selection rule is clearly monotone, and the paymen ts are the threshold bids. Th us, w e can assume without loss of generalit y that bids and costs coincide. In the follo wing, we analyze the fruga lit y ratio of E V , and sho w that E V is comp etitive. Lemma 3.1 E V has frugality r atio at most α . Pro of. W e start b y considering only cost v ectors with exactly one non-zero en try , i.e., of the form c = c v · 1 v . F or suc h a cost v ector, the mec hanism will choose a subset of V \ { v } as 10 the winning set, a nd pay each u in t ha t subset her threshold bid. W e calculate the threshold bids of all these agen ts u . First, consider any ag en t u ∼ v . If u w ere to raise her bid ab o ve ( q u /q v ) · c v , while all agen ts b esides u and v con tinued to bid 0, then the set V \ { u } w ould b e c heap er than { u } with resp ect to the new bid v ector c ′ . Therefore, u w ould not b e part of the winning v ertex co v er. Th us u ’s threshold payme nt is at most ( q u /q v ) · c v . Next, consider any agen t u 6∼ v . Because V \ { u, v } is a vertex cov er, u cannot raise her bid ab ov e zero without losing, so her threshold bid is 0. Hence, the total pay ment of E V is at most P ( c ) = (1 /q v ) · c v · P u ∼ v q u . On the other hand, by the deﬁnition of ν v and linearit y of ν , w e ha ve that ν ( c ) = c v ν v , so the frugality ratio for cost v ectors o f the form c v · 1 v is (1 /q v ) · c v · P u ∼ v q u c v ν v = 1 q v · X u ∼ v 1 ν v · q u = 1 q v · α · q v = α, where the second equalit y fo llo w ed b ecause the vector q is a n eigenv ector of K with eigen v a lue α . Th us, fo r an y cost v ector with only one non-zero en try , the frugality ratio is at most α . No w consider an arbitrary cost ve ctor c , and write it as c = P v c v 1 v . W e claim that P ( c ) ≤ P v c v P ( 1 v ). F or consider any v ertex u ∈ S winning with cost v ector c . If the cost v ector we re c v 1 v instead, u ’s paymen t w ould be ( q u /q v ) · c v if u ∼ v and 0 otherwise. On the other hand, when the cost v ector is c , if u bids strictly more than P v ∼ u q u /q v · c v , then u cannot b e in the winning set, as replacing u with all its neigh b ors w ould give a c heap er solution with resp ect to t he costs c ′ . Th us, eac h no de u gets paid at most P v ∼ u q u /q v · c v with cost v ector c , and the total pay ment is at most P ( c ) = X u X v ∼ u q u q v · c v = X v c v · X u ∼ v q u q v = X v c v P ( 1 v ) . On the other hand, w e hav e that ν ( c ) ≥ X v c v ν ( 1 v ) = X v c v ν v , b ecause of the fo llo wing a rgumen t: for eac h v , let x ( v ) b e a an optimal solution fo r the LP (2) with cost v ector 1 v . Then, simply by linearit y , the v ector x = P v c v x ( v ) is feasible for ( 2 ) with cost ve ctor c , and a chiev es the sum o f the pa ymen ts. Th us, the optimal solution t o (2) with cost v ector c can hav e no smaller t o tal pa ymen ts. Com bining the results o f the previous t w o paragraphs, w e hav e the fo llowing b ound on the frugalit y ra t io: max c P ( c ) ν ( c ) ≤ max c P v c v P ( 1 v ) P v c v ν v ≤ max v P ( 1 v ) ν v ≤ α . Next, w e prov e that no other mec hanism can do asymptotically b etter. Lemma 3.2 L et M b e any truthful vertex c o v e r me chani sm on G . Then, M has frugality r atio at le ast α 2 . 11 Pro of. W e construct a directed graph G ′ = ( V , E ′ ) from G b y directing each edge e of G in a t least one direction. Consider any edge e = ( u, v ) of G . Let c b e the cost v ector in whic h c u = q u , c v = q v , a nd c i = 0 for all i 6 = u, v . When M is run on the cost/bid v ector c , at least one o f u and v mu st b e in the winning set S ; otherwise, it w ould not b e a v ertex co v er. If u ∈ S , then add the directed edge ( v , u ) to E ′ . Similarly , if v ∈ S , then add ( u , v ) to E ′ . If b oth u, v ∈ S , then add b oth directed edges. By doing this for all edges e ∈ G , w e ev en tually obtain a graph G ′ . No w giv e eac h no de v a w eight q v . Each no de- weigh ted directed graph ( V , E ′ ) con ta ins at least one no de v suc h that X u :( v, u ) ∈ E ′ q u ≥ X u :( u,v ) ∈ E ′ q u , (see, e.g., the pro of of Lemma 11 in [18]), and hence X u :( v, u ) ∈ E ′ q u ≥ 1 2 X u : u ∼ v q u . Fix an y suc h no de v in G ′ with respect to the w eigh ts q v . No w consider the cost vec to r c with c v = q v and c i = 0 for all i 6 = v . By monotonicity of the selection rule o f M (whic h follo ws from the t r ut hf ulness of M ), at least a ll no des u suc h that ( v , u ) ∈ G ′ m ust b e part of the selected set S of M , and must b e paid at least q u . Therefore, the total pa ymen t of M is at least X u :( v, u ) ∈ G ′ q u ≥ 1 2 X u ∼ v q u = 1 2 ν v X u ∼ v 1 ν v q u = 1 2 ν v · α q v , where the last equalit y fo llow ed from the fact that q is an eigen v ector of the matrix K . On the other hand, as in the pro of of Lemma 3.1, ν ( c ) = ν v q v for our cost vector c , so the frugalit y ra t io is at least 1 2 α , when the cost v ector is c . Com bining Lemma 3 .1 and Lemma 3.2, w e ha ve pro v ed the follo wing theorem: Theorem 3.3 E V is 2 -c om p etitive for V e rtex Cover auctions. Remark 3.4 The low er b ound of 1 2 α on the frugality ratio of any mec hanism can p oten tially b e la r ge. F or instance, for a complete bipartite graph K n,n , we hav e α = Θ( n ). Th us, suc h large o ve rpaymen ts are inheren t in truthful mec hanisms in general. Ho we ver, truthful mec hanisms may b e m uch more frugal on sp eciﬁc classes of graphs. Remark 3.5 E V in general do es not run in p olynomial time. F o r the ﬁnal step, computing a minim um-cost v ertex co ver with resp ect to the scaled costs, we could use a monot o ne 2-approx imat ion, as suggested by Elkind et al. [11]. The hardness of computing K is more sev ere. How ev er, notice that for sp eciﬁc classes of graphs, suc h as degree-b ounded o r tria ngle- free graphs, K can b e computed eﬃcien tly , giving us non-trivial p olynomial- time mec hanisms for V ertex Cov er on those classes. This issue is discuss ed more in Section 6. 12 3.2 Nash Equilibria and the F ractional Clique Problem In this section, w e sho w that the Nash Equilibrium v alues ν v used for scaling of the matrix actually hav e a natura l in terpretation. T o state the result, recall that the fr actional clique numb er is the solution to the linear program Maximize P u x u sub ject to P u ∈ I x u ≤ 1 for all independen t sets I x u ≥ 0 for all u (4) The fr ac tion a l chr omatic numb er is the solution of the dual problem, where we hav e a v ariable y I for each indep enden t set I and a constraint P I ∋ u y I ≥ 1 for eac h vertex u , and w e minimize P I y I . By LP duality , the fra ctional clique n um b er and the fr actional c hro ma t ic num b er a r e equal. Prop osition 3.6 L et G v b e the sub gr aph in d uc e d by the neighb o rh o o d of v but without v itself. Then, ν v is exactly the fr actional c l i q ue numb er, and thus the fr actional chr omatic numb er, of G v . Pro of. Let x b e any bid vec to r feasible for the LP (2). First, fo r all v ertices u that do not share an edge with v , we mus t hav e x u = 0, b ecause V \ { u, v } is a feasible set. So w e can restrict our atten tion to G v . F or a set I , w e write x ( I ) = P u ∈ I x u for the total bids of the v ertices in I . If I is an indep enden t set in G v , then x ( I ) ≤ 1. The reason is t ha t the set V \ I is also feasible, and w ould cost less than V \ { v } if x ( I ) exceeded 1. Th us, an y feasible bid v ector x induces a feasible solution to the LP (4), of the same total cost. Con v ersely , if we hav e a feasible solution to the LP (4), w e can extend it to a bid v ector for all agents b y setting x v = 1, and x u = 0 for all v ertices u outside v ’s neighborho o d. W e need to sho w that eac h feasible set T , i.e., eac h v ertex cov er, has tota l bid x ( T ) at least a s large as the set V \ { v } . If T do es not contain v , it m ust con tain a ll of v ’s neigh b ors; it th us has the same bid as V \ { v } b y deﬁnition. Otherwise, b ecause V \ T is an indep enden t set, the feasibilit y of x f or the LP (4) implies that x ( V \ T ) ≤ 1. Thu s, x ( T ) ≥ x ( V ) − 1 = x ( V \ { v } ), and the t wo LPs 2 and (4) ha ve the same v alue. Standard ra ndomized rounding arg uments (see, e.g., [22 ]) imply that for an y graph, t he c hromatic n um b er and the fractional chromatic n umber are within a facto r O (log n ) of eac h other. Therefore, an y appro ximation hardness results for G raph Coloring also apply to t he F ractiona l Clique Problem with at most a lo ss of logarithmic fa ctor s. In particular, the result of F eige and Kilian [13] implies that unless ZPP=NP , ν v cannot b e appro ximated to within a factor O ( n 1 − ǫ ) in p olynomial time, for an y ǫ > 0 . 13 3.3 Comp osabilit y and a G eneral Design Approac h V ertex Co v er auctions can b e used naturally as a wa y to deal with other ty p es of set systems. First, pre-pro cess the set system by remo ving a subset of agen ts, turning the remaining set system in to a V ertex Co v er instance; then, run E V on that instance. The impo rtan t part is to c ho ose t he pre-pro cessing rule to ensure that the o verall mech - anism is b oth truthful and comp etitive. A condition termed c omp osabi l i ty in [1, Deﬁnition 5.2] is suﬃcien t to ensure truthfulness. W e sho w here t ha t a comparison b et wee n lo w er b ounds is suﬃcien t t o show comp etitiv eness. Deﬁnition 3.7 (Comp osabilit y [1]) L et σ b e a sele ction rule mappin g bid ve ctors to s ub- sets of (r emaining) agents. We say that σ is comp osable if σ ( b ) = T im plies that σ ( b ′ e , b − e ) = T for any e ∈ T and b ′ e ≤ b e . In other wor ds, not only c an a winning agent not b e c om e a loser by biddi n g lowe r; she c annot e ven ch a n ge whic h s et c ontaining her wins. F ormally , when we talk ab out “remo ving” a set of agen ts from a set system, w e are replacing ( E , F ) with ( T , F | T ), where T = σ ( b ), and F | T := { S ∈ F | S ⊆ T } . Theorem 3.8 L et σ b e a c omp osable sele ction rule with the fol lowing additional pr op- erty: F or al l monop oly-fr e e se t systems ( E , F ) in the class, and al l c ost ve ctors c , writing ( E ′ , F ′ ) := ( σ ( c ) , F | σ ( c ) ) : 1. ( E ′ , F ′ ) is a V ertex Cover instanc e , and 2. ν ( E ′ , F ′ ) ( c ) ≤ κ · ν ( E , F ) ( c ) . L et the Remo ve-Co ver Mec hanism RC M c onsist of running E V on ( E ′ , F ′ ) . T hen RC M is a truthful 2 κ -c omp etitive me chani s m. Pro of. T ruthfulness is pro ved in [1, Lemma 5.3 ]. The pro of is short, and w e include a v ersion here for completeness. Consider an y agent e , and a bid ve ctor b − e for a g en ts other than e . Because σ is comp osable, and thus also monotone, there is a threshold bid τ e suc h that e wins iﬀ her bid is at most τ e . F urthermore, whenev er b e ≤ τ e , the set σ ( b ) is uniquely determined, a nd indep endent of b e . Thus , whenev er b e ≤ τ e , E V will b e run on the same set system ( σ ( b ) , F | σ ( b ) ), a nd t he selection rule of E V on this set system is monoto ne. Hence, the o verall selection rule of RC M is monotone fo r e , implying directly that RC M is truthful. The upp er b ound on the frugality rat io of RC M follows simply fro m Lemma 3.1 a nd the assumption of the theorem: P RC M ( c ) ≤ α (( E ′ , F ′ )) · ν ( E ′ , F ′ ) ( c ) ≤ α (( E ′ , F ′ )) · κ · ν ( E , F ) ( c ) . T o pro v e the low er b ound, let M b e an y truthful mec hanism for ( E , F ), and let ( E ′ , F ′ ) b e the V ertex Co ve r set system maximizing α (( E ′ , F ′ )). W e consider cost v ectors c with c e = ∞ (or some v ery la rge ﬁnite v alues) for e / ∈ E ′ . F or suc h cost v ectors, w e can safely disregard all elemen ts e / ∈ E ′ altogether, as they will not aﬀect the solutions to the LP (2), nor b e part of an y solution selected b y M . 14 But then, M is exactly a mec hanism selecting a feasible solution to the V ertex Co v er in- stance ( E ′ , F ′ ). By Lemma 3.2, M thus has frugality ratio at least α (( E ′ , F ′ )) / 2, completing the pro of. A simple general wa y to obtain a comp osable rule is to choose the set with the minimum total cost, from some subset of the feasible sets: Lemma 3.9 L et σ b e any rule with c onsistent tie br e ak i n g sele cting a set S minim i z ing b ( S ) over al l se ts S with a c e rtain pr op erty P . Then σ is c omp osable. Pro of. Consider any agent e who is part of the winning set S with respect to b . If e ’s bid decreases b y ǫ , the cost of S decreases by ǫ , while the costs of all other sets decrease b y at most ǫ . Th us, b ecause t ies are broken consisten tly , S will still b e selecte d. 4 A Mec hanism for Flo w s W e apply the metho do lo gy of Theorem 3.8 to design a mec hanism F M for purc hasing k edge-disjoin t s - t pa t hs. W e are giv en a (directed) gra ph G = ( V , E ), source s , sink t , and target num b er k . As discussed earlier, the agen ts are edges of G . W e assume that G is monop oly-free, which is equiv alen t to saying t ha t the minim um s - t cut con tains a t least k + 1 edges. F or conv enience, w e will refer to a set of k edge-disjoin t s - t paths simply as a k -ﬂow, and omit s and t . T o sp ecify F M , all w e need to do is describ e a comp osable pre-pro cessing rule σ . Our rule is simple: Cho o se ( k + 1) edge-disjoin t s - t paths, of minim um to t al bid with resp ect to b , breaking ties lexicographically . W e call suc h a subgraph a ( k + 1) - ﬂow , where it is implicit that w e are only intere sted in integer ﬂo ws, and iden tify t he ﬂow with its edge set. Call the minim um-cost ( k + 1)-ﬂow H . (In Section 3.3, w e generically referred to this set system as ( E ′ , F ′ ).) Theorem 4.1 The me chanism F M is truthful and 2( k + 1) -c omp etitive and runs in p oly- nomial time. W e show this theorem in three parts. First, w e establish t ha t the k -ﬂow problem on H indeed fo r ms a V ertex Co v er instance (Lemma 4.2). By fa r the most diﬃcult step is showing that the lo w er b ound satisﬁes ν H ( c ) ≤ ( k + 1 ) · ν G ( c ) for all cost v ectors c (L emma 4.4). The comp osabilit y of σ f o llo ws fro m Lemma 3.9. T ogether, these three facts allo w us to apply Theorem 3.8, and conclude that F M is a trut hf ul 2 ( k + 1)-comp etitiv e mec hanism. Finally , w e ve rif y that F M runs in p olynomial time (Lemma 4.7 ). Lemma 4.2 The instanc e ( E ′ , F ′ ) whose fe asible sets ar e al l k -ﬂows on H is a V ertex Cover set system. 15 Pro of. Recall that H is a minimal ( k + 1)-ﬂow, a fact that w e exploit rep eatedly in this pro of. The edges of H are the v ertices in the V ertex Co ve r instance. F or clarit y , consider explicitly the graph R , whic h con ta ins a v ertex u e for eac h edge e ∈ H , and an edge b et w een u e , u e ′ if and only if remov ing e w ould create a monop oly for e ′ . This is the case iﬀ there exists at least one minim um s - t cut in H con taining b oth e a nd e ′ ; in particular, R is symmetric. The construction is illustrated f or the case k = 2 in Figure 1. An alternativ e c har acterization of the edges in R is giv en in Prop osition 4.3 b elow , a nd will b e used as part of this pro o f. F or any set of edges E ′ in H , let N ( E ′ ) b e the correspo nding set o f no des in R . Thus , for an y minim um s - t cut E ′ , the set N ( E ′ ) forms a clique in R . s t u v w x y u v w x y Figure 1: A minimally 3-connected graph (left) a nd the resulting ve r t ex cov er instance for k = 2 (rig h t). If E ′ is a k -ﬂow, then for an y pair o f edges e, e ′ that lie on a minim um s - t cut, E ′ m ust con tain at least o ne of e, e ′ . Thus , N ( E ′ ) is a v ertex cov er of R . Con v ersely , let E ′ b e a set of edges in H suc h that N ( E ′ ) is a ve rtex cov er of R . W e will sho w that f o r ev ery s - t cut F ⊆ E , a t least k edges of E ′ cross F , i.e., | E ′ ∩ F | ≥ k . This will imply that E ′ is a k -ﬂow. Assume for contradiction that | E ′ ∩ F | < k . Because N ( E ′ ) is a verte x co v er of R , there can b e no edge betw een an y pair of v ertices in N ( F \ E ′ ) in R . By deﬁnition, this means that for any pair e, e ′ ∈ F \ E ′ , there is no minim um s - t cut con taining b oth e and e ′ . By Prop osition 4.3 b elo w, this is equiv alen t to sa ying that for each pair e, e ′ ∈ F \ E ′ , the g raph H con tains a path from e to e ′ or a pa th from e ′ to e . Consider a directed graph whose v ertices are the edges F \ E ′ , with an edge from e to e ′ whenev er H con t a ins a path from e to e ′ . By the ab o ve argument, this graph is a t o urnamen t graph, and thus contains a Ha milto nian path. That is, there is an ordering e 1 , . . . , e ℓ of the edges in F \ E ′ suc h that eac h e i +1 is reac hable fro m e i in H . By adding a pa th from s to e 1 and from e ℓ to t , we thu s obta in an s - t path P containing all edges in F \ E ′ . The graph H \ P is a k -ﬂo w, so the set E ′ ∩ F , hav ing size less than k , cannot b e an s - t cut in H \ P . Let P ′ b e an s - t path in H \ P disjoint from E ′ ∩ F . By construction, P ′ is also disjoint from F \ E ′ . Th us, w e ha ve found an s - t path P ′ in H disjoint from F , contradicting the assumption that F is an s - t cut. Prop osition 4.3 L et H b e a gr aph c onsisting of k + 1 e dge-disj o int s - t p aths, a nd let e = ( u, v ) , e ′ = ( u ′ , v ′ ) b e two e dges of H . Then, ther e is a min imum s - t cut c ontaining b oth e and e ′ if and on ly if ther e is no p ath fr om v to u ′ and no p ath fr om v ′ to u . Pro of. Assume that there is a path from v to u ′ . Let P b e a concatenation of an s - v path using e , the pat h from v to u ′ , and a pa th fr o m u ′ to t using e ′ . Then, H \ P is a k -ﬂow , 16 and therefore has k edge-disjoin t paths. Any s - t cut in H m ust thus contain at least k edges from H \ P , and no s - t cut with few er than k + 2 edges can con ta in b oth e and e ′ . Con v ersely , if there is no minimum cut con taining b oth e, e ′ , then ev ery minim um cut in H \ { e, e ′ } m ust contain k edges. Thus, H \ { e, e ′ } con tains k edge-disjoin t s - t paths. Remo ving these paths from H leav es us with a 1- ﬂow, i.e., one s - t path. By construction, this path m ust contain e and e ′ ; th us, at least one is reac hable f rom the other. Lemma 4.4 ν H ( c ) ≤ ( k + 1 ) · ν G ( c ) for al l c . Pro of. Let S b e the c heap est k -ﬂo w in G with resp ect to t he costs c . Bec ause H is a ( k + 1)-ﬂow, Corollary 4.6 b elo w implies that ν H ( c ) = k · π H ( c ), where π H ( c ) is the cost of the most expensiv e s - t pa t h in H . Let x b e a solution to the LP (2) with cost v ector c on the graph G . Deﬁne a graph G ′ consisting of all edges in S , a s w ell as all edges that are in at least one tight feasible set T (i.e., a set T for whic h the constrain t (iii) is tigh t, meaning tha t x ( T ) = x ( S )). By deﬁnition, G ′ con tains at least k + 1 edge-disjoint s - t paths. Lemma 4.5 b elow (the k ey step) implies that all s - t paths in G ′ ha v e the same tot a l bid with resp ect to x . Let P b e an s - t path in G ′ of maximum total cost c ( P ). By individual rationality (Constrain t (i) in the LP (2)), we hav e tha t x ( P ) ≥ c ( P ), and hence x ( P ′ ) ≥ c ( P ) for all s - t paths P ′ . In particular, ν G ( c ) = x ( S ) ≥ k · c ( P ). Since H is a minim um-cost ( k + 1)-ﬂow , and b ecause G ′ con tains at least k + 1 edge-disjoin t s - t paths, w e hav e π H ( c ) ≤ ( k + 1) c ( P ). Thus , ν G ( c ) ≥ k · c ( P ) ≥ k k + 1 · π H ( c ) = 1 k + 1 · ν H ( c ) , whic h completes the pro of. Lemma 4.5 L et x b e a solution to the LP (2) , and G ′ as in the pr o of of L emm a 4.4. L e t v b e an arbitr ary no de in G , and P 1 , P 2 two p aths fr om v to t . Th e n, x ( P 1 ) = x ( P 2 ) . Pro of. Let E b e the collection of all t ig h t k -ﬂo ws from s t o t except S , i.e., the set of all F suc h that F 6 = S , F consists o f exactly k edge-disjoin t s - t paths, and x ( F ) = x ( S ). W e deﬁne a directed multigraph ˜ G as follo ws: for eac h F ∈ E , w e add to ˜ G a copy of eac h edge e ∈ F (creating duplicate copies of edges e whic h are in multiple ﬂo ws F ). W e call these edges forwar d e dges . In addition, f or eac h edge e = ( u , v ) ∈ S , w e add |E | copies of the b ack w ar d e dge ( v , u ) to ˜ G , i.e., w e direct e the other w ay . In the resulting m ultigraph, eac h no de v has an in-degree equal to its out- degree. F or v 6 = s, t this follo ws since eac h edge set we added constitutes a ﬂow. F or v = s, t , it f o llo ws since each F ∈ E adds k edges out of s and into t , while the | E | copies o f S add k |E | edges in to s and out of t . As a result, ˜ G is Eulerian, a fact w e use b elow . W e deﬁne a mapping γ ( e ), whic h assigns to eac h edge e ∈ ˜ G its “original” edge in G . As usual, w e extend notat io n and write γ ( R ) = { γ ( e ) | e ∈ R } for an y set R of edges. W e will b e particularly in terested in analyzing collections of cycle s in ˜ G . W e sa y that t w o cycles C 1 , C 2 are image-d i s joint if γ ( C 1 ) ∩ γ ( C 2 ) = ∅ . A cycle set is an y set of zero o r 17 more image-disjoint cyc les in ˜ G (whic h w e iden tify with its edge set), and Γ denotes the collection of all cycle sets. F or a cycle set C ∈ Γ, let C → and C ← denote the set of forward and backw ard edges in C , resp ectiv ely . Then, w e deﬁne φ ( C ) = S ∪ γ ( C → ) \ γ ( C ← ). It is easy to see that for eac h cycle set C , φ ( C ) is a k -ﬂow in G ′ . Conv ersely , f or ev ery k - ﬂo w F in G ′ , there is a cycle set C ∈ Γ with φ ( C ) = F . W e assign eac h edge e ∈ ˜ G a w eight w e . F or forward edges e , w e set w e = x γ ( e ) , while for back ward edges e = ( v , u ), w e set w e = − x γ ( e ) . Notice that b ecause each cop y of S con tributes weigh t − x ( S ), and each set F ∈ E con tributes x ( F ) = x ( S ), t he sum of all w eigh ts in ˜ G is 0. No w, let C b e an y cycle set, and F = φ ( C ) its corresp o nding k -ﬂow. W e ha v e X e ∈C w e = x ( F \ S ) − x ( S \ F ) = x ( F ) − x ( S ) , Th us F is tight, i.e., x ( F ) = x ( S ), if and only if P e ∈C w e = 0. W e next sho w that for an y cycle C in ˜ G , the k -ﬂo w φ ( C ) is tigh t, a nd therefore that C has total w eigh t zero, i.e., P e ∈ C w e = 0. Assume for con tradiction that this is not the case, and let C b e a cycle with P e ∈ C w e 6 = 0. Let F = φ ( C ) b e the corr esp o nding k -ﬂow. Because w e sho w ed ab o ve that P e ∈ C w e = x ( F ) − x ( S ), we can rule out the p o ssibilit y that P e ∈ C w e < 0; otherwise, x ( F ) < x ( S ), whic h w ould violate Constrain t (iii) o f the LP (2). If P e ∈ C w e > 0, consider the m ultigraph ˜ G ′ obtained by remov ing C fro m ˜ G . Its total w eigh t is P e / ∈ C w e < 0 , b ecause the sum of all w eigh ts in ˜ G is 0 (as sho wn ab o ve). Since ˜ G is Eulerian, so is ˜ G ′ , and its edges can b e partitioned in to a collection of edge-disjoin t cycles { C 1 , . . . , C ℓ } . By the Pigeonhole Principle, a t least one of the C i m ust hav e negative total w eight. But then x ( F i ) < x ( S ), where F i = φ ( C i ), violating Constrain t (iii) of (2) as in the previous case. This completes the pro of that φ ( C ) is tight for an y cycle C . By our observ ation ab o ve , the total w eight of a n y cycle C is zero. Finally , w e pro ve the statemen t o f the lemma by induction on a rev erse top olog ical sorting of the v ertices v —that is, an ordering in whic h the index o f v is a t least a s large as t he index of any u suc h that ( v , u ) ∈ G ′ . Because G ′ is acyclic, suc h a sorting exists. The base case v = t is trivial. F o r v 6 = t , let P 1 , P 2 b e tw o v - t paths. W e distinguish three cases, based on the ﬁrst edges e 1 = ( v , u 1 ) , e 2 = ( v , u 2 ) of t he pat hs P 1 , P 2 . 1. If ˜ G contains a forw a rd edge ( v , u 1 ) and a backw ard edge ( u 2 , v ) (o r vice v ersa), then since ev ery set of edges a dded to ˜ G is a ﬂo w, ˜ G m ust contain a v - t path P ′ 1 en tirely consisting of forw ard edges and starting with e 1 , and a t - v path P ′ 2 en tirely consisting of bac kw ar d edges and ending with e 2 (bac kw ard). Applying the induction hypothesis to u 1 and u 2 , since P 1 and P ′ 1 share their ﬁrst edges and similarly for P 2 and P ′ 2 , w e ha v e x ( γ ( P ′ 1 )) = x ( P 1 ) a nd x ( γ ( P ′ 2 )) = x ( P 2 ). Because P ′ 1 ∪ P ′ 2 forms a cycle, it has total w eigh t 0. Then x ( γ ( P ′ 2 )) = − w P ′ 2 = w P ′ 1 = x ( γ ( P ′ 1 )), and so x ( P 1 ) = x ( P 2 ). 2. If ˜ G con ta ins forward edges ( v , u 1 ) and ( v , u 2 ), t hen it con ta ins v - t paths P ′ 1 , P ′ 2 starting with ( v , u 1 ) and ( v , u 2 ), resp ectiv ely , and consisting en tirely of forward edges. Applying 18 the induction h yp othesis to u 1 and u 2 , w e hav e x ( γ ( P ′ 1 )) = x ( P 1 ) and x ( γ ( P ′ 2 )) = x ( P 2 ). Since ev ery set of edges added to ˜ G is a ﬂow, ˜ G m ust contain an s - v path P consisting en tirely of forw ard edges, and ˜ G m ust also contain a t - s path P ′ consisting en t ir ely of bac kw ard edges. Becaus e P ∪ P ′ ∪ P ′ i forms a cycle f or eac h i ∈ { 1 , 2 } and has total we ig ht zero, w e obtain x ( P i ) = x ( γ ( P ′ i )) = − w P ∪ P ′ for each i . In particular, x ( P 1 ) = x ( P 2 ). 3. Finally , if ˜ G con tains bac kw ar d edges ( u 1 , v ) and ( u 2 , v ), w e apply an argument similar to the previous case. By induction, x ( γ ( P ′ 1 )) = x ( P 1 ), and x ( γ ( P ′ 2 )) = x ( P 2 ). Again using the fact that ˜ G consists o f ﬂows , it con tains t - v paths P ′ 1 , P ′ 2 with resp ectiv e last edges ( u 1 , v ) and ( u 2 , v ). In addition, ˜ G contains a v - s path P consisting en tirely o f bac kw ard edges, and an s - v path P ′ consisting entirely o f forward edges. Then for eac h i ∈ { 1 , 2 } , P ∪ P ′ ∪ P ′ i forms a cycle with total w eigh t zero, so x ( P 1 ) = x ( P 2 ). As a corollary , w e can deriv e a c hara cterization of Nash Equilibria in ( k + 1)-ﬂows . Corollary 4.6 If G is a ( k + 1) -ﬂow, then a bid ve ctor x is a Nash Equilibrium if and only if x ( P ) = π G ( c ) for al l s - t p a ths P . In p articular, al l Nash Equilibria have the same total c ost x ( S ) = k · π G ( c ) , wh er e S is the winn ing set. Pro of. First, b ecause G is a ( k + 1)- ﬂo w, the graph G ′ deﬁned in the pro of of Lemma 4.4 actually equals G , since it must contain k + 1 edge-disjoin t s - t paths. If x is a Nash Equi- librium, then b y Lemma 4.5, all s - t paths P ha v e the same total bid x ( P ). Let ˆ P b e an s - t path maximizing c ( P ), i.e., c ( ˆ P ) = π G ( c ). G \ ˆ P is a k - ﬂo w, and clearly the c heap est k -ﬂow b y deﬁnition of ˆ P . Therefore, all agen ts in ˆ P lose, and x ( ˆ P ) = c ( ˆ P ) by Constraint (ii) of the LP (2). Finally , we sho w that the mec hanism E V runs in p olynomial time for the special case of graphs deriv ed from k -ﬂows . Lemma 4.7 F or the V ertex Cover in stanc e derive d fr om c omputing a k -ﬂow on a ( k + 1) - ﬂow, the me cha nism E V runs in p olynomial time. Pro of. There a r e t wo steps whic h are o f concern: computing the v alues ν v , and ﬁnding the c heap est ve rt ex cov er with resp ect to the scaled bids. The latter is exactly a Minim um Cost Flo w problem by Prop osition 4.2, and th us solv able in p olynomial t ime with standard algorithms [2]. F or the former, w e claim that ν u e = k for all u e ∈ R . By Prop o sition 3.6 and LP duality , ν u e is upp er b ounded b y the c hromatic num b er of u e ’s neigh b orho o d, and low er b ounded by its clique n um b er. Since eac h edge e ∈ H is part of a minim um cut of size k + 1, and the edges of t he minim um cut form a clique in R , the clique num b er of u e ’s neigh b orho o d is at least k . O n the other hand, w e can decomp ose H into k + 1 edge-disjoint paths, and color the v ertices corresp onding to eac h pat h with its o wn color in R . By Prop osition 4.3, this is a v alid coloring, and sho ws that u e ’s neigh b orho o d is k -colorable. 19 Remark 4.8 The f actor 2 in the result of Theorem 4.1 comes fro m the fa ctor 1 2 in the low er b ound in Lemma 3.2. Using a more reﬁned lo wer b ound ba sed on Y oung’s Inequality , Chen et al. [7] sho w ed that for an unscaled v ersion of the V ertex Cov er mec hanism, the f actor 1 2 in the lo we r b ound is unnecessary . F or the instances of V ertex Co v er pro duced as a result of the pruning in this section, t he mec hanism from [7] coincides with E V , and hence F M is the same as the ﬂo w mec hanim [7]. Chen et al. also show ed tha t while F M is ( k + 1)-comp etitiv e when compared a gainst the buy er-optimal lo we r b ound [18], it is in fact optimal compared to the buyer-pessimal v ersion [11]. 5 A Mec hanism for C u ts As a second application of our metho dology , w e giv e a competitive mec hanism C M for purc hasing an s - t cut, giv en a (directed) graph G = ( V , E ), source s , and sink t . Again, the agen ts are edges. Here, the necessary monop oly-freeness is equiv alent to G not containing the edge ( s, t ). As b efore, it suﬃces to sp ecify and a na lyze a comp osable pre-pro cessing rule σ . Our pre- pro cessing rule is to compute a minimum - cost set E ′ of edges (with resp ect to the submitted bids b ), suc h that E ′ con tains at least tw o edges from each s - t path. W e call suc h an edge set a double cut . W e show b elow restricting the set system to E ′ giv es a V ertex Co v er instance, and at most increases the cost of the winning set b y a factor of 2. 5.1 Restricting to a double cut T o restrict the set system to E ′ , w e contract a ll edges in E \ E ′ . Since no suc h edge will b e cut, con tra cting it ensures that its endp oin ts will alwa ys lie on the same side of the cut. Let H denote the resulting graph. W e b egin with a simple structural lemma ab out H. Lemma 5.1 In H, al l s - t p aths hav e l e ngth exactly 2. Pro of. If there w ere an s - t path of length 1 in H, i.e., a n edge ( s, t ), then consider the edge ( u, v ) in the origina l graph corr esp o nding to ( s, t ). Because u was con tra cted with s , and v with t , there m ust b e an s - u path and a v - t pat h in G using only edges from E \ E ′ . In that case, ( u, v ) is the only edge on this path contained in E ′ , so E ′ cannot ha v e b een a double cut. Similarly , if there we re an s - t path P of length at least 3, then at least one edge ( u, v ) of P has neither s nor t as an endpo in t. This edge could b e safely con tracted, i.e., remo v ed from E ′ , in whic h case E ′ w as not a minim um-cost double cut. Theorem 5.2 The double cut sele ction rule is c omp osable and pr o duc es a V ertex Cover instanc e with ν H ( c ) ≤ 2 ν G ( c ) . F urthermor e, b oth the sele ction rule and the subse quent V ertex Cover me chanism c an b e c o m pute d in p olynomial time. Thus, C M is a p olynomial- time 4-c om p etitive m e cha n ism. 20 Comp osabilit y follow s fro m Lemma 3.9, and the ﬁnal conclusion then follo ws f rom Theorem 3.8 once w e establish the other claims. W e can obtain a V ertex Co ver instance b y imp osing a graph structure on H, treating eac h edge as a ve rtex a nd adding an edge ( e, e ′ ) b et we en a n y t wo edges that form an s - t path. A set of edges is a n s - t cut if a nd only if it contains at least one of e, e ′ in eac h suc h pair, so it is a v ertex cov er of the resulting graph. W e can think of this in t urn as a ﬂo w problem as follows. Lemma 5.1 implies that H is of the following form: in addition to s and t , t here are v ertices v 1 , . . . , v ℓ , and fo r eac h i = 1 , . . . , ℓ , a set of pa r a llel edges E i from s to v i , and a set of parallel edges E ′ i from v i to t . Any s - t cut has to include, for eac h i , all of E i or a ll of E ′ i . Th us, if w e deﬁne a minimally 2-connected graph consisting of a series of v ertices u 0 , u 1 , . . . , u ℓ with tw o v ertex-disjoin t paths o f length | E i | and | E ′ i | b et w een u i − 1 and u i for each i , an s - t cut in H is a 1-ﬂow from u 0 to u ℓ . W e can then apply Lemma 4.2. Notice that this equiv alence also establishes that E V r uns in p o lynomial time on the instances pro duced by this selection r ule. As b efore, the k ey part is to analyze the increase in the low er b ound. Lemma 5.3 F or al l c os t ve ctors c , ν H ( c ) ≤ 2 ν G ( c ) . Pro of. Let ( S, S ) b e the c heap est s - t cut in G with resp ect to the costs c , and x a solution to the LP (2) with cost v ector c on the gr a ph G . Let C b e the set of all minim um s - t cuts ( T , T ) with resp ect to the costs x ; th us, eac h o f these cuts has cost x ( E ( S, S )). Deﬁne T − = T ( T , T ) ∈C T , and T + = S ( T , T ) ∈C T . Then, b oth ( T − , T − ) a nd ( T + , T + ) a re minim um s - t cuts as well (see, e.g., [2, Exercis e 6.39]) . F urthermore, the edge sets E ( T − , T − ) and E ( T + , T + ) are disjoint. F or assume tha t there is an edge e = ( u, v ) in common b et wee n these sets. Then, u ∈ T ( T , T ) ∈C T and v ∈ T ( T , T ) ∈C T . In particular, this implies that u ∈ S a nd v ∈ S . As stated ab o ve in equation (3), since x maximizes the LP (2 ) , Constrain t (iii) m ust b e tigh t for some f easible set excluding e , since otherwise the bid x e could b e increased. Let ( T , T ) b e the corresp onding cut. Then x ( E ( T , T )) = x ( E ( S, S )) , a nd e do es not cross ( T , T ). Th us, either b o th u and v are in T , or b oth are in T . Since ( T , T ) ∈ C , this give s a con tradiction. No w deﬁne G ′ := E ( T − , T − ) ∪ E ( T + , T + ). Because G ′ consists o f tw o disjoin t s - t cuts, it is a double cut, and the cost- minimality of H implies tha t c ( G ′ ) ≥ c ( H ) . These t w o cuts b oth hav e minimal cost, so ν G ( c ) = x ( G ′ ) / 2. By the “individual rationality” L P constrain t (i), x ( G ′ ) ≥ c ( G ′ ), and hence ν G ( c ) = x ( G ′ ) 2 ≥ c ( G ′ ) 2 ≥ c ( H ) 2 ≥ ν H ( c ) 2 . F or the last inequalit y , notice that in the “Nash Equilibrium” o n H, for each i , the c heap er of E i and E ′ i will collectiv ely raise their bids to the cost of the more exp ensiv e one, so the total bid of the winning set will b e ν H ( c ) = P i max( c ( E i ) , c ( E ′ i )) ≤ c ( H ) . 21 5.2 A P rimal-Dual Algorithm for Minim um Doub le-Cuts Finally , w e presen t a p olynomial time alg orithm to compute a minimum - cost double cut. The minim um-cost double cut is c haracterized by in teger solutions to the follow ing LP , where P denotes the set of all s - t paths in G . Minimize P e ∈ E c e x e sub ject to P e ∈ P x e ≥ 2 for all P ∈ P x e ≤ 1 for all edges e ∈ E x e ≥ 0 for all e, (5) Remark 5.4 It is no t diﬃcult to sho w that the constrain t matrix for this LP is totally unimo dular. By a w ell-kno wn theorem [27], b ecause the right-hand sides of the constrain ts are in tegral, tot a l unimo dularity implies that a ll the v ertices of the L P’s p olytop e a r e in tegra l. Since there is a separation ora cle for the LP (as w ell as an equiv alen t p olynomial-sized LP form ulatio n), a n integer solution can b e found in p o lynomial time, g iving us a p olynomial- time algorithm. How ev er, the resulting algorithm is ra t her ineﬃcien t. Here, w e presen t a more eﬃcien t prima l- dual alg o rithm generalizing the F o rd-F ulk erson Max-Flo w algo rithm. The dual of the LP is Maximize 2 P P ∈P f P − P e r e sub ject to P P : e ∈ P f P ≤ c e + r e for all e ∈ E f P , r e ≥ 0 for all P ∈ P and all e ∈ E . (6) W e interpret the dual v ariables f P as describing a ﬂow in the usual w a y . That is, the ﬂo w along eac h edge e is f e = X P : e ∈ P f P . W e sa y that e is saturated if f e = c e + r e . W e call r e the r elief o n e : in order to send more ﬂo w on an edge e , w e can increase its capacit y , but w e pa y for it in the o b jectiv e function. It is w o r t h augmen ting the ﬂow along a path so long as at most one edge o n the path is saturated, since increasing the ﬂo w and the relief of the saturated edge at the same time increases the dual ob jectiv e. Our primal-dual algorithm is similar to the F o r d- F ulkers on algorithm, and is based on the same concept of a residual graph. The residual graph con tains forw a r d e dges for a ll edges e in the o riginal graph, even when they ar e satur ate d , b ecause it is p ossible t o send more ﬂo w b y adding relief. In addition, if e = ( u, v ) carries ﬂo w f e , then the residual graph, as usual, con ta ins the b ackwar d e d ge ( v , u ) with capacity f e . T o capture how muc h r elief w ould ha v e to b e added to augmen t the ﬂo w along a path, we deﬁne, for eac h edge e in the residual graph, a length ℓ e as follo ws: 1. If e is a saturated forw ard edge, then ℓ e = 1. 22 2. If ( v , u ) is a bac kw ard edge suc h that ( u, v ) has p ositive relief, then ℓ ( v,u ) = − 1. 3. The lengths of all other edges are ℓ e = 0. F or paths P , w e deﬁne ℓ ( P ) = P e ∈ P ℓ e . W e give our primal-dual algorithm as Algorithm 1. Algorithm 1 Flo w computation for Minim um Double Cut 1: Flo w Computation: 2: Let f b e an arbitrary maxim um ﬂow on G . 3: while there is an s - t path P with ℓ ( P ) ≤ 1 in the residual g r aph G f do 4: Let P b e suc h a path with minimum length ℓ ( P ) . 5: Augmen t the ﬂow on P b y δ , while sim ultaneously increasing the relief of any saturated edge b y δ , and decreasing the relief of any bac kward edge by δ , for the smallest v alue of δ suc h that this action increases ℓ ( P ), i.e., the smallest δ such that either a new forw ard edge b ecomes saturated or the relief on a bac kw ard edge b ecomes zero. Notice tha t f or any path P of length at most 1, augmen ting t he ﬂow increases the dual ob jectiv e. This follow s since the total n um b er of saturated edges is at most one great er than the total num b er of bac kw ard edges with relief; for the latter, each unit of ﬂow r educes the total relief by one unit, while for the former, eac h unit of ﬂo w increases the to t a l relief b y one unit. Th us, the to tal increase in relief for sending δ units of ﬂo w is at most δ , while the ﬁrst term of the ob jectiv e function, i.e., the v alue of the ﬂo w, increases b y 2 δ . As with the F o rd-F ulk erson alg orithm, the running time could b e pseudo-p olynomial with a p o or choice of the augmen ting path P . But breaking ties for the smallest total n um b er of edges in P g iv es strongly p olynomial running time, as with the Edmonds-Karp algorithm. When the algorithm terminates, w e hav e a set T of saturated edges, and a subset R ⊆ T of r elief e dges e with r e > 0. W e will pic k t w o edge-disjoin t s - t cuts ( S 1 , S 1 ), ( S 2 , S 2 ) with the prop erties that: 1. Only saturated edges cross either of the cuts. 2. S 1 ⊂ S 2 . 3. Eac h relief edge crosses one of the t wo chose n cuts. This will nat ur a lly satisfy all complemen tary slack ness conditions for the t wo LPs, and th us pro ve opt ima lity o f the cuts. T o deﬁne and compute the t wo cuts, w e fo cus on the graph G ′ obtained from the residual graph b y remo ving all fo r ward edges e with f e = 0. Impo rtan tly , we use the same notion of length deﬁned ab ov e. F rom now on, all reference s to reac hability , distances, etc. a r e with resp ect to G ′ . F or each no de v , let d v denote the minim um distance from s to v in G ′ . W e sho w next that G ′ has no negativ e cycles, so these distances are well-deﬁn ed. Lemma 5.5 G ′ has no p ath fr om the s i n k t to the sour c e s of length strictly less than − 1 . In p articular, G ′ c ontain s no ne gative cycles. 23 Pro of. W e show b y induction tha t these prop erties hold for the residual graph in eac h iteration. Since G ′ is obtained from the residual graph only b y deleting edges with zero ﬂow (and th us length 0 or 1 only), distances can only increase in G ′ . Initially , all edges ha ve length 0 or 1, and there are no bac kward edges, so the claim clearly holds. If the residual graph con tained a negativ e-length cycle C , then C w ould ha v e to contain at least o ne ﬂow-carrying edge e = ( u, v ). Since e has incoming ﬂow from s and outgoing ﬂow to t , the residual gra ph would contain a pa t h of bac kw ard edges from u to s and one f r om t to v . Th us, an y nega t ive cycle w ould giv e arbitrarily negative - length paths from t to s . It is therefore enough to establish the ﬁrst claim. Consider an iteration when ﬂo w is a ugmen ted along a path P . Supp ose that this generates a t - s path P ′ in the residual graph of length strictly less than − 1. P ∪ P ′ giv es a cycle. If w e assign edges in P their length prior to t he augmen tation, and edges in P ′ their length after t he augmen tat ion, then the total length of the cycle P ∪ P ′ is negativ e. The only edges in P ′ whose length can ha v e decreased through the augmen t a tion are the bac kw ard v ersions of edges to whic h relief w as added b y P . They we re saturated b efor e the augmen tation, so their forw ard length was 1, a nd t heir ba ckw ard length aft er augmen t a tion is − 1. No w consider remo ving edges tha t app ear b oth forw ard and bac kw ard in P ∪ P ′ . W e obtain a union o f edge-disjoin t cycles, suc h that all edges in these cycles w ere presen t in the residual graph prio r to the ﬂow aug men tation. Of these edge-disjoin t cycles, b y the argumen t of the previous para graph, at least one cycle C has negat ive length with respect to the previous par a graph’s deﬁnition. The edges in P ′ \ P don’t change their length, so C had negativ e length b efore the augmen tation, contradicting the induction h yp othesis. F or tw o no des u, v , we write u → 0 v if there is a path o f length at most 0 fro m u to v in G ′ . W e now deﬁne the cuts. Let S 1 := { v | d v ≤ 0 } . D eﬁne E ′ := { ( u, v ) ∈ R | d u > 0 } . No w, let U b e the set of all v ertices lying on a v - t path f or some edge ( u, v ) ∈ E ′ , and let S 2 b e the set of all ve rtices y suc h that y → 0 w for some w ∈ U . Clearly , ( S 1 , S 1 ) and ( S 2 , S 2 ) deﬁne t wo s - t cuts using only saturated edges. Lemma 5.6 No e dge cr osses b oth cuts ( S 1 , S 1 ) and ( S 2 , S 2 ) . Each r elief e dge e ∈ R cr osses one of the cuts ( S 1 , S 1 ) or ( S 2 , S 2 ) . Pro of. T o pro v e the ﬁrst claim, supp ose that e = ( u , v ) crosses b oth cuts. By the deﬁnition of S 2 , that means that there is an edge e ′ = ( u ′ , v ′ ) ∈ E ′ ⊆ R suc h that there is a path of length at most 0 f rom v to some no de w on a v ′ - t path. Consider the path from s to u (of length at most 0), f o llo we d by e (of length at most 1), follow ed b y the path from v to w , and then the path to v ′ bac kw ards, follo we d by e ′ bac kw ards. This is a path of length at most 0 from s to u ′ , meaning that u ′ should hav e b een in S 1 , and con tradicting that e ′ w as in E ′ . T o pro v e the second claim, supp ose that a relief edge e = ( u, v ) crosses neither o f the cuts. W e distinguish tw o cases: 1. If u ∈ S 1 then, since e do es not cross either cut, v ∈ S 1 , so d v ≤ 0. But then the s - v path of length at most 0, follow ed by e bac kw ards (of length − 1), follow ed b y the u - s path bac kw ard (of length at most 0) giv es a negative cycle, con tradicting Lemma 5.5. 24 2. If u / ∈ S 1 , then d u > 0 a nd e ∈ E ′ . Thus v ∈ U , a nd since v → 0 v , we ha ve v ∈ S 2 . Since e do es not cross either cut, we also hav e u ∈ S 2 . This means t ha t t here is a w ∈ U and e ′ = ( u ′ , v ′ ) ∈ E ′ suc h that w lies on a v ′ - t path, and u → 0 w . Now consider the path from t to v ( o f length at most 0), then using e bac kw ar ds (of length − 1), then the length-0 path from u to w and the path f rom w t o v backw ards (of length at most 0), fo llow ed by e ′ bac kw ards, and the path from u ′ to s ba ckw ards (o f length at most 0). This gives a t - s path of total length at most − 2, aga in con tradicting L emma 5.5. Lemma 5.6 implies that the set R of relief edges forms a double cut. Thus our algorithm ﬁnds a minim um-cost double cut in p olynomial time. Remark 5.7 As in Remark 4.8 for the ﬂow mech a nism F M , w e can sho w that on instances deriv ed fr om the pruning step, E V coincides with the mec hanism of [7 ]. Thus , the tigh ter analysis sho ws that C M is in fact 2-comp etitiv e. W e conjecture that C M is indeed o ptimal when compared against the buy er-p essimal low er b o und o f [11]. 6 Directio ns for F uture W ork W e ha ve presen ted nov el truthful and comp etitive mec hanisms for three imp ortant com bi- natorial problems: V ertex Cov ers, k - ﬂo ws, and s - t cuts. The V ertex Co ve r mec hanism w as based on scaling the submitted bids by multipliers derived as comp onen ts of the dominan t eigen v ector of a suitable matrix. Bo t h the ﬂo w and cut mec hanisms w ere based on pruning the input graph, and then applying the V ertex Cov er mec hanism to the pruned v ersion. Be- sides the individual mec hanisms, we b elieve that the metho dology o f reducing input instances to V ertex Cov er pro blems ma y b e of intere st for future frugal mec hanism design. In general, the V ertex Co ve r mec hanism do es not run in p olynomial time, due to t wo obstacles. First, computing the matrix K requires computing the largest fractio na l clique size in t he neigh b orho o d of each no de v . Subsequen tly , computing the solution with resp ect to scaled costs requires ﬁnding a c heap est v ertex co v er. F or the second o bstacle, it seems quite lik ely that mono t one algorithms suc h as the one in [11] could b e adapted to our setting, and yield constan t- factor approximations. Ho w ev er, the diﬃcult y of computing the en tries of K seems more sev ere. In fact, w e conjecture that no p olynomial- time truthful mec hanism fo r V ertex Co v er can b e constant-competitiv e. This result w o uld b e quite in teresting, in that it w ould sho w that the r equiremen ts of incen tive-compatibilit y and computational tra ctabilit y together can lead to signiﬁcan tly w orse g ua ran tees tha n either requiremen t alo ne. A p ositive resolution o f this conjecture w ould thus b e akin to the t yp es of hardness results demonstrated recen tly f or the Com binatorial Public Pro ject Problem [26]. While our metho dolog y o f designing comp osable pre-pro cessing algorithms will lik ely b e useful for ot her problems as w ell, it do es not apply to all set systems. It is fairly easy to construct set systems for which no suc h pruning alg orithm is p ossible. Ev en when pruning is p ossible in principle, it ma y come with a large blo wup in costs. 25 Th us, the following bigger ques tion still stands: whic h classes of set systems admit constan t-comp etitive mec hanisms? The main obstacle is our inability to pro ve strong lo w er b ounds on frugalit y rat ios. T o date, all lo we r b ounds (here, as w ell as in [12, 18]) are based on pairwise comparisons b etw een agen ts, whic h can then b e used to sho w that certain agen ts, b y virtue of losing, will cause large pa ymen ts. This techn ique was exactly the motiv ation for our V ertex Co v er approac h. In order to mov e b ey ond V ertex Cov er based mec hanisms, it will b e necessary to explore lo w er b ound tec hniques b ey ond the one used in this pap er. In recen t joint w ork with the authors of [7], w e hav e sho wn that the factor 1 2 in the lo w er b ound of Lemma 3.2 can b e remo v ed, th us sho wing that E V is optimal. The pro of o f this result will b e presen ted in a joint f ull v ersion of the presen t pap er with [7]. Ac kno wledgmen t W e w ould like to thank Edith Elkind, Uriel F eige, Nick Gravin, Anna Karlin, T ami T amir, and Mihalis Y annak akis for useful discussions and p ointers, and anony mous review ers fo r useful feedbac k. D.K. is support ed in part b y an NSF CAREER Aw ard, an ONR Y oung In v estigator Aw ard and a n a w ard from the Sloan F oundation. C.M. is supp o rted in part b y the McDonnell F oundation. References [1] Gagan Aggarwal and Jason D . Hartline. 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Frugal and Truthful Auctions for Vertex Covers, Flows, and Cuts

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