Frugality Ratios And Improved Truthful Mechanisms for Vertex Cover

Frugality Ratios And Improved Truthful Mehanisms for Vertex Cover Edith Elkind Hebrew University of Jerusalem, Israel, and University of Southampton, Southampton, SO17 1BJ, U.K. Leslie Ann Goldberg University of Liverpool Liverpool L69 3BX, U.K. Paul Goldberg University of Liverpool Liverpool L69 3BX, U.K. ABSTRACT In set-system autions, there are several overlapping teams of agents, and a task that an be ompleted by any of these teams. The autioneer s goal is to hire a team and pay as little as possible. Examples of this setting inlude shortest-path autions and vertex-over autions. Reently, Karlin, Kempe and Tamir introdued a new definition of frugality ratio for this problem. Informally, the frugality ratio is the ratio of the total payment of a mehanism to a desired payment bound. The ratio aptures the extent to whih the mehanism overpays, relative to pereived fair ost in a truthful aution. In this paper, we propose a new truthful polynomial-time aution for the vertex over problem and bound its frugality ratio. We show that the solution quality is with a onstant fator of optimal and the frugality ratio is within a onstant fator of the best possible worst-ase bound; this is the first aution for this problem to have these properties. Moreover, we show how to transform any truthful aution into a frugal one while preserving the approximation ratio. Also, we onsider two natural modifiations of the definition of Karlin et al., and we analyse the properties of the resulting payment bounds, suh as monotoniity, omputational hardness, and robustness with respet to the draw-resolution rule. We study the relationships between the different payment bounds, both for general set systems and for speifi set-system autions, suh as path autions and vertex-over autions. We use these new definitions in the proof of our main result for vertex-over autions via a bootstrapping tehnique, whih may be of independent interest. Categories and Subjet Desriptors F.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity; J.4 [Computer Appliations]: Soial and Behavioral Sienes eonomis General Terms Algorithms, Eonomis, Theory This researh is supported by the EPSRC researh grants Algorithmis of Network-sharing Games and Disontinuous Behaviour in the Complexity of randomized Algorithms. Keywords Autions, Frugality, Vertex Cover 1. INTRODUCTION In a set system aution there is a single buyer and many vendors that an provide various servies. It is assumed that the buyer s requirements an be satisfied by various subsets of the vendors; these subsets are alled the feasible sets. A widely-studied lass of setsystem autions is path autions, where eah vendor is able to sell aess to a link in a network, and the feasible sets are those sets whose links ontain a path from a given soure to a given destination; the study of these autions has been initiated in the seminal paper by Nisan and Ronen [19] (see also [1, 10, 9, 6, 15, 7, 20]). We assume that eah vendor has a ost of providing his servies, but submits a possibly larger bid to the autioneer. Based on these bids, the autioneer selets a feasible subset of vendors, and makes payments to the vendors in this subset. Eah seleted vendor enjoys a profit of payment minus ost. Vendors want to maximise profit, while the buyer wants to minimise the amount he pays. A natural goal in this setting is to design a truthful aution, in whih vendors have an inentive to bid their true ost. This an be ahieved by paying eah seleted vendor a premium above her bid in suh a way that the vendor has no inentive to overbid. An interesting question in mehanism design is how muh the autioneer will have to overpay in order to ensure truthful bids. In the ontext of path autions this topi was first addressed by Arher and Tardos [1]. They define the frugality ratio of a mehanism as the ratio between its total payment and the ost of the heapest path disjoint from the path seleted by the mehanism. They show that, for a large lass of truthful mehanisms for this problem, the frugality ratio is as large as the number of edges in the shortest path. Talwar [21] extends this definition of frugality ratio to general set systems, and studies the frugality ratio of the lassial VCG mehanism [22, 4, 14] for many speifi set systems, suh as minimum spanning trees and set overs. While the definition of frugality ratio proposed by [1] is wellmotivated and has been instrumental in studying truthful mehanisms for set systems, it is not ompletely satisfatory. Consider, for example, the graph of Figure 1 with the osts AB = BC = Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page. To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speifi permission and/or a fee. EC 07, June 13 16, 2007, San Diego, California, USA. Copyright 2007 ACM 978-1-59593-653-0/07/0006...$5.00. A B C Figure 1: The diamond graph D

CD = 0, AC = BD = 1. This graph is 2-onneted and the VCG payment to the winning path ABCD is bounded. However, the graph ontains no A D path that is disjoint from ABCD, and hene the frugality ratio of VCG on this graph remains undefined. At the same time, there is no monopoly, that is, there is no vendor that appears in all feasible sets. In autions for other types of set systems, the requirement that there exist a feasible solution disjoint from the seleted one is even more severe: for example, for vertex-over autions (where vendors orrespond to the verties of some underlying graph, and the feasible sets are vertex overs) the requirement means that the graph must be bipartite. To deal with this problem, Karlin et al. [16] suggest a better benhmark, whih is defined for any monopoly-free set system. This quantity, whih they denote by ν, intuitively orresponds to the value of a heapest Nash equilibrium. Based on this new definition, the authors onstrut new mehanisms for the shortest path problem and show that the overpayment of these mehanisms is within a onstant fator of optimal. 1.1 Our results Vertex over autions We propose a truthful polynomial-time aution for vertex over that outputs a solution whose ost is within a fator of 2 of optimal, and whose frugality ratio is at most 2, where is the maximum degree of the graph (Theorem 4). We omplement this result by proving (Theorem 5) that for any and n, there are graphs of maximum degree and size Θ(n) for whih any truthful mehanism has frugality ratio at least /2. This means that the solution quality of our aution is with a fator of 2 of optimal and the frugality ratio is within a fator of 4 of the best possible bound for worst-ase inputs. To the best of our knowledge, this is the first aution for this problem that enjoys these properties. Moreover, we show how to transform any truthful mehanism for the vertex-over problem into a frugal one while preserving the approximation ratio. Frugality ratios Our vertex over results naturally suggest two modifiations of the definition of ν in [16]. These modifiations an be made independently of eah other, resulting in four different payment bounds TUmax, TUmin, NTUmax, and NTUmin, where NTUmin is equal to the original payment bound ν of in [16]. All four payment bounds arise as Nash equilibria of ertain games (see the full version of this paper [8]); the differenes between them an be seen as the prie of initiative and the prie of ooperation (see Setion 3). While our main result about vertex over autions (Theorem 4) is with respet to NTUmin = ν, we make use of the new definitions by first omparing the payment of our mehanism to a weaker bound NTUmax, and then bootstrapping from this result to obtain the desired bound. Inspired by this appliation, we embark on a further study of these payment bounds. Our results here are as follows: 1. We observe (Proposition 1) that the four payment bounds always obey a partiular order that is independent of the hoie of the set system and the ost vetor, namely, TUmin NTUmin NTUmax TUmax. We provide examples (Proposition 5 and Corollaries 1 and 2) showing that for the vertex over problem any two onseutive bounds an differ by a fator of n 2, where n is the number of agents. We then show (Theorem 2) that this separation is almost best possible for general set systems by proving that for any set system TUmax/TUmin n. In ontrast, we demonstrate (Theorem 3) that for path autions TUmax/TUmin 2. We provide examples (Propositions 2, 3 and 4) showing that this bound is tight. We see this as an argument for the study of vertexover autions, as they appear to be more representative of the general team -seletion problem than the widely studied path autions. 2. We show (Theorem 1) that for any set system, if there is a ost vetor for whih TUmin and NTUmin differ by a fator of α, there is another ost vetor that separates NTUmin and NTUmax by the same fator and vie versa; the same is true for the pairs (NTUmin, NTUmax) and (NTUmax, TUmax). This symmetry is quite surprising, sine, e.g., TUmin and NTUmax are obtained from NTUmin by two very different transformations. This observation suggests that the four payment bounds should be studied in a unified framework; moreover, it leads us to believe that the bootstrapping tehnique of Theorem 4 may have other appliations. 3. We evaluate the payment bounds introdued here with respet to a heklist of desirable features. In partiular, we note that the payment bound ν = NTUmin of [16] exhibits some ounterintuitive properties, suh as nonmonotoniity with respet to adding a new feasible set (Proposition 7), and is NP-hard to ompute (Theorem 6), while some of the other payment bounds do not suffer from these problems. This an be seen as an argument in favour of using weaker but effiiently omputable bounds NTUmax and TUmax. Related work Vertex-over autions have been studied in the past by Talwar [21] and Calinesu [5]. Both of these papers are based on the definition of frugality ratio used in [1]; as mentioned before, this means that their results only apply to bipartite graphs. Talwar [21] shows that the frugality ratio of VCG is at most. However, sine finding the heapest vertex over is an NP-hard problem, the VCG mehanism is omputationally infeasible. The first (and, to the best of our knowledge, only) paper to investigate polynomial-time truthful mehanisms for vertex over is [5]. This paper studies an aution that is based on the greedy alloation algorithm, whih has an approximation ratio of log n. While the main fous of [5] is the more general set over problem, the results of [5] imply a frugality ratio of 2 2 for vertex over. Our results improve on those of [21] as our mehanism is polynomial-time omputable, as well as on those of [5], as our mehanism has a better approximation ratio, and we prove a stronger bound on the frugality ratio; moreover, this bound also applies to the mehanism of [5]. 2. PRELIMINARIES In most of this paper, we disuss autions for set systems. A set system is a pair (E, F), where E is the ground set, E = n, and F is a olletion of feasible sets, whih are subsets of E. Two partiular types of set systems are of interest to us shortest path systems, in whih the ground set onsists of all edges of a network, and the feasible sets are paths between two speified verties s and t, and vertex over systems, in whih the elements of the ground set are the verties of a graph, and the feasible sets are vertex overs of this graph. In set system autions, eah element e of the ground set is owned by an independent agent and has an assoiated non-negative ost e. The goal of the entre is to selet (purhase) a feasible set. Eah element e in the seleted set inurs a ost of e. The elements that are not seleted inur no osts. The aution proeeds as follows: all elements of the ground set make their bids, the entre selets a feasible set based on the bids and makes payments to the agents. Formally, an aution is defined by an alloation rule A : R n F and a payment rule P : R n R n. The alloation rule takes as input a vetor of bids and deides whih of the sets in F should be seleted. The payment rule also takes as input a vetor of bids and deides how muh to pay to eah agent. The standard requirements are individual rationality, i.e., the payment to eah agent should be at least as high as his inurred ost (0 for agents not in the seleted set and e for agents in the

seleted set) and inentive ompatibility, or truthfulness, i.e., eah agent s dominant strategy is to bid his true ost. An alloation rule is monotone if an agent annot inrease his hane of getting seleted by raising his bid. Formally, for any bid vetor b and any e E, if e A(b) then e A(b 1,..., b e,..., b n) for any b e > b e. Given a monotone alloation rule A and a bid vetor b, the threshold bid t e of an agent e A(b) is the highest bid of this agent that still wins the aution, given that the bids of other partiipants remain the same. Formally, t e = sup{b e R e A(b 1,..., b e,..., b n)}. It is well known (see, e.g. [19, 13]) that any aution that has a monotone alloation rule and pays eah agent his threshold bid is truthful; onversely, any truthful aution has a monotone alloation rule. The VCG mehanism is a truthful mehanism that maximises the soial welfare and pays 0 to the losing agents. For set system autions, this simply means piking a heapest feasible set, paying eah agent in the seleted set his threshold bid, and paying 0 to all other agents. Note, however, that the VCG mehanism may be diffiult to implement, sine finding a heapest feasible set may be intratable. If U is a set of agents, (U) denotes P w U w. Similarly, b(u) denotes P w U bw. 3. FRUGALITY RATIOS We start by reproduing the definition of the quantity ν from [16, Definition 4]. Let (E, F) be a set system and let S be a heapest feasible set with respet to the true osts e. Then ν(, S) is the solution to the following optimisation problem. Minimise B = P e S be subjet to (1) b e e for all e E (2) P e S\T be P e T \S e for all T F (3) for every e S, there is a T e F suh that e T e and P e S\T e b e = P e T e\s e The bound ν(, S) an be seen as an outome of a two-stage proess, where first eah agent e S makes a bid b e stating how muh it wants to be paid, and then the entre deides whether to aept these bids. The behaviour of both parties is affeted by the following onsiderations. From the entre s point of view, the set S must remain the most attrative hoie, i.e., it must be among the heapest feasible sets under the new osts e = e for e S, e = b e for e S (ondition (2)). The reason for that is that if (2) is violated for some set T, the entre would prefer T to S. On the other hand, no agent would agree to a payment that does not over his osts (ondition (1)), and moreover, eah agent tries to maximise his profit by bidding as high as possible, i.e., none of the agents an inrease his bid without violating ondition (2) (ondition (3)). The entre wants to minimise the total payout, so ν(, S) orresponds to the best possible outome from the entre s point of view. This definition aptures many important aspets of our intuition about fair payments. However, it an be modified in two ways, both of whih are still quite natural, but result in different payment bounds. First, we an onsider the worst rather than the best possible outome for the entre. That is, we an onsider the maximum total payment that the agents an extrat by jointly seleting their bids subjet to (1), (2), and (3). Suh a bound orresponds to maximising B subjet to (1), (2), and (3) rather than minimising it. If it is the agents who make the original bids (rather than the entre), this kind of bidding behaviour is plausible. On the other hand, in a game in whih the entre proposes payments to the agents in S and the agents aept them as long as (1), (2) and (3) are satisfied, we would be likely to observe a total payment of ν(, S). Hene, the differene between these two definitions an be seen as the prie of initiative. Seond, the agents may be able to make payments to eah other. In this ase, if they an extrat more money from the entre by agreeing on a vetor of bids that violates individual rationality (i.e., ondition (1)) for some bidders, they might be willing to do so, as the agents who are paid below their osts will be ompensated by other members of the group. The bids must still be realisti, i.e., they have to satisfy b e 0. The resulting hange in payments an be seen as the prie of o-operation and orresponds to replaing ondition (1) with the following weaker ondition (1 ): b e 0 for all e E. (1 ) By onsidering all possible ombinations of these modifiations, we obtain four different payment bounds, namely TUmin(, S), whih is the solution to the optimisation problem Minimise B subjet to (1 ), (2), and (3). TUmax(, S), whih is the solution to the optimisation problem Maximise B subjet to (1 ), (2), and (3). NTUmin(, S), whih is the solution to the optimisation problem Minimise B subjet to (1), (2), and (3). NTUmax(, S), whih is the solution to the optimisation problem Maximise B subjet to (1), (2), (3). The abbreviations TU and NTU orrespond, respetively, to transferable utility and non-transferable utility, i.e., the agents ability/inability to make payments to eah other. For onreteness, we will take TUmin() to be TUmin(, S) where S is the lexiographially least amongst the heapest feasible sets. We define TUmax(), NTUmin(), NTUmax() and ν() similarly, though we will see in Setion 6.3 that, in fat, NTUmin(, S) and NTUmax(, S) are independent of the hoie of S. Note that the quantity ν() from [16] is NTUmin(). The seond modifiation (transferable utility) is more intuitively appealing in the ontext of the maximisation problem, as both assume some degree of o-operation between the agents. While the seond modifiation an be made without the first, the resulting payment bound TUmin(, S) is too strong to be a realisti benhmark, at least for general set systems. In partiular, it an be smaller than the total ost of the heapest feasible set S (see Setion 6). Nevertheless, we provide the definition as well as some results about TUmin(, S) in the paper, both for ompleteness and beause we believe that it may help to understand whih properties of the payment bounds are important for our proofs. Another possibility would be to introdue an additional onstraint P P e S be e S e in the definition of TUmin(, S) (note that this ondition holds automatially for TUmax(, S), as TUmax(, S) NTUmax(, S)); however, suh a definition would have no diret game-theoreti interpretation, and some of our results (in partiular, the ones in Setion 4) would no longer be true. REMARK 1. For the payment bounds that are derived from maximisation problems, (i.e., TUmax(, S) and NTUmax(, S)), onstraints of type (3) are redundant and an be dropped. Hene, TUmax(, S) and NTUmax(, S) are solutions to linear programs, and therefore an be omputed in polynomial time as long as we have a separation orale for onstraints in (2). In ontrast,

NTUmin(, S) an be NP-hard to ompute even if the size of F is polynomial (see Setion 6). The first and third inequalities in the following observation follow from the fat that ondition (1 ) is stritly weaker than ondition (1). PROPOSITION 1. TUmin(, S) NTUmin(, S) NTUmax(, S) TUmax(, S). Let M be a truthful mehanism for (E, F). Let p M() denote the total payments of M when the atual osts are. A frugality ratio of M with respet to a payment bound is the ratio between the payment of M and this payment bound. In partiular, φ TUmin(M) = sup p M()/TUmin(), φ TUmax(M) = sup p M()/TUmax(), φ NTUmin(M) = sup p M()/NTUmin(), φ NTUmax(M) = sup p M()/NTUmax(). We onlude this setion by showing that there exist set systems and respetive ost vetors for whih all four payment bounds are different. In the next setion, we quantify this differene, both for general set systems, and for speifi types of set systems, suh as path autions or vertex over autions. EXAMPLE 1. Consider the shortest-path aution on the graph of Figure 1. The heapest feasible sets are all paths from A to D. It an be verified, using the reasoning of Propositions 2 and 3 below, that for the ost vetor AB = CD = 2, BC = 1, AC = BD = 5, we have TUmax() = 10 (with b AB = b CD = 5, b BC = 0), NTUmax() = 9 (with b AB = b CD = 4, b BC = 1), NTUmin() = 7 (with b AB = b CD = 2, b BC = 3), TUmin() = 5 (with b AB = b CD = 0, b BC = 5). 4. COMPARING PAYMENT BOUNDS 4.1 Path autions We start by showing that for path autions any two onseutive payment bounds an differ by at least a fator of 2. PROPOSITION 2. There is an instane of the shortest-path problem for whih we have NTUmax()/NTUmin() 2. PROOF. This onstrution is due to David Kempe [17]. Consider the graph of Figure 1 with the edge osts AB = BC = CD = 0, AC = BD = 1. Under these osts, ABCD is the heapest path. The inequalities in (2) are b AB + b BC AC = 1, b BC + b CD BD = 1. By ondition (3), both of these inequalities must be tight (the former one is the only inequality involving b AB, and the latter one is the only inequality involving b CD). The inequalities in (1) are b AB 0, b BC 0, b CD 0. Now, if the goal is to maximise b AB + b BC + b CD, the best hoie is b AB = b CD = 1, b BC = 0, so NTUmax() = 2. On the other hand, if the goal is to minimise b AB + b BC + b CD, one should set b AB = b CD = 0, b BC = 1, so NTUmin() = 1. PROPOSITION 3. There is an instane of the shortest-path problem for whih we have TUmax()/NTUmax() 2. PROOF. Again, onsider the graph of Figure 1. Let the edge osts be AB = CD = 0, BC = 1, AC = BD = 1. ABCD is the lexiographially-least heapest path, so we an assume that S = {AB, BC,CD}. The inequalities in (2) are the same as in the previous example, and by the same argument both of them are, in fat, equalities. The inequalities in (1) are b AB 0, b BC 1, b CD 0. Our goal is to maximise b AB + b BC + b CD. If we have to respet the inequalities in (1), we have to set b AB = b CD = 0, b BC = 1, so NTUmax() = 1. Otherwise, we an set b AB = b CD = 1, b BC = 0, so TUmax() 2. PROPOSITION 4. There is an instane of the shortest-path problem for whih we have NTUmin()/TUmin() 2. PROOF. This onstrution is also based on the graph of Figure 1. The edge osts are AB = CD = 1, BC = 0, AC = BD = 1. ABCD is the lexiographially least heapest path, so we an assume that S = {AB, BC, CD}. Again, the inequalities in (2) are the same, and both are, in fat, equalities. The inequalities in (1) are b AB 1, b BC 0, b CD 1. Our goal is to minimise b AB + b BC +b CD. If we have to respet the inequalities in (1), we have to set b AB = b CD = 1, b BC = 0, so NTUmin() = 2. Otherwise, we an set b AB = b CD = 0, b BC = 1, so TUmin() 1. In Setion 4.4 (Theorem 3), we show that the separation results in Propositions 2, 3, and 4 are optimal. 4.2 Connetions between separation results The separation results for path autions are obtained on the same graph using very similar ost vetors. It turns out that this is not oinidental. Namely, we an prove the following theorem. THEOREM 1. For any set system (E, F), and any feasible set S, max max TUmax(, S) NTUmax(, S) = max NTUmax(, S) NTUmin(, S), NTUmax(, S) NTUmin(, S) = max NTUmin(, S) TUmin(, S), where the maximum is over all ost vetors for whih S is a heapest feasible set. The proof of the theorem follows diretly from the four lemmas proved below; more preisely, the first equality in Theorem 1 is obtained by ombining Lemmas 1 and 2, and the seond equality is obtained by ombining Lemmas 3 and 4. We prove Lemma 1 here; the proofs of Lemmas 2 4 are similar and an be found in the full version of this paper [8]. LEMMA 1. Suppose that is a ost vetor for (E, F) suh that S is a heapest feasible set and TUmax(, S)/NTUmax(, S) = α. Then there is a ost vetor suh that S is a heapest feasible set and NTUmax(, S)/NTUmin(, S) α. PROOF. Suppose that TUmax(, S) = X and NTUmax(, S) = Y where X/Y = α. Assume without loss of generality that S onsists of elements 1,..., k, and let b 1 = (b 1 1,..., b 1 k) and b 2 = (b 2 1,..., b 2 k) be the bid vetors that orrespond to TUmax(, S) and NTUmax(, S), respetively. Construt the ost vetor by setting i = i for i S, i = min{ i, b 1 i } for i S. Clearly, S is a heapest set under. Moreover, as the osts of elements outside of S remained the same, the right-hand sides of all onstraints in (2) did not hange, so any bid vetor that satisfies (2) and (3) with respet to, also satisfies them with respet to. We will onstrut two bid vetors b 3 and b 4 that satisfy onditions (1), (2), and (3) for the ost vetor, and

X 4 X 5 P i j \ P P i j+2 \ P X 3 X 6 X 0 e i j e i j+1 e i j+2 x i j x i j+1 y ij x i j+2 y y i j+1 i j+2 X 2 X 1 P i j+1 \ P Figure 2: Graph that separates payment bounds for vertex over, n = 7 have P i S b3 i = X, P i S b4 i = Y. As NTUmax(, S) X and NTUmin(, S) Y, this implies the lemma. We an set b 3 i = b 1 i : this bid vetor satisfies onditions (2) and (3) sine b 1 does, and we have b 1 i min{ i, b 1 i } = i, whih means that b 3 satisfies ondition (1). Furthermore, we an set b 4 i = b 2 i. Again, b 4 satisfies onditions (2) and (3) sine b 2 does, and sine b 2 satisfies ondition (1), we have b 2 i i i, whih means that b 4 satisfies ondition (1). LEMMA 2. Suppose is a ost vetor for (E, F) suh that S is a heapest feasible set and NTUmax(, S)/NTUmin(, S) = α. Then there is a ost vetor suh that S is a heapest feasible set and TUmax(, S)/NTUmax(, S) α. LEMMA 3. Suppose that is a ost vetor for (E, F) suh that S is a heapest feasible set and NTUmax(, S)/NTUmin(, S) = α. Then there is a ost vetor suh that S is a heapest feasible set and NTUmin(, S)/TUmin(, S) α. LEMMA 4. Suppose that is a ost vetor for (E, F) suh that S is a heapest feasible set and NTUmin(, S)/TUmin(, S) = α. Then there is a ost vetor suh that S is a heapest feasible set and NTUmax(, S)/NTUmin(, S) α. 4.3 Vertex-over autions In ontrast to the ase of path autions, for vertex-over autions the gap between NTUmin() and NTUmax() (and hene between NTUmax() and TUmax(), and between TUmin() and NTUmin()) an be proportional to the size of the graph. PROPOSITION 5. For any n 3, there is a an n-vertex graph and a ost vetor for whih TUmax()/NTUmax() n 2. PROOF. The underlying graph onsists of an (n 1)-lique on the verties X 1,..., X n 1, and an extra vertex X 0 adjaent to X n 1. The osts are X1 = X2 = = Xn 2 = 0, X0 = Xn 1 = 1. We an assume that S = {X 0, X 1,..., X n 2} (this is the lexiographially first vertex over of ost 1). For this set system, the onstraints in (2) are b Xi + b X0 Xn 1 = 1 for i = 1,..., n 2. Clearly, we an satisfy onditions (2) and (3) by setting b Xi = 1 for i = 1,..., n 2, b X0 = 0. Hene, TUmax() n 2. For NTUmax(), there is an additional onstraint b X0 1, so the best we an do is to set b Xi = 0 for i = 1,..., n 2, b X0 = 1, whih implies NTUmax() = 1. Combining Proposition 5 with Lemmas 1 and 3, we derive the following orollaries. COROLLARY 1. For any n 3, we an onstrut an instane of the vertex over problem on a graph of size n that satisfies NTUmax()/NTUmin() n 2. COROLLARY 2. For any n 3, we an onstrut an instane of the vertex over problem on a graph of size n that satisfies NTUmin()/TUmin() n 2. Figure 3: Proof of Theorem 3: onstraints for ˆP ij and ˆP ij+2 do not overlap 4.4 Upper bounds It turns out that the lower bound proved in the previous subsetion is almost tight. More preisely, the following theorem shows that no two payment bounds an differ by more than a fator of n; moreover, this is the ase not just for the vertex over problem, but for general set systems. We bound the gap between TUmax() and TUmin(). Sine TUmin() NTUmin() NTUmax() TUmax(), this bound applies to any pair of payment bounds. THEOREM 2. For any set system (E, F) and any ost vetor, we have TUmax()/TUmin() n. PROOF. Assume wlog that the winning set S onsists of elements 1,..., k. Let 1,..., k be the true osts of elements in S, let b 1,..., b k be their bids that orrespond to TUmin(), and let b 1,..., b k be their bids that orrespond to TUmax(). Consider the onditions (2) and (3) for S. One an pik a subset L of at most k inequalities in (2) so that for eah i = 1,..., k there is at least one inequality in L that is tight for b i. Suppose that the jth inequality in L is of the form b i1 + + b it (T j \ S). For b i, all inequalities in L are, in fat, equalities. Hene, by adding up all of them we obtain k P i=1,...,k b i P j=1,...,k (Tj \ S). On the other hand, all these inequalities appear in ondition (2), so they must hold for b i, i.e., P i=1,...,k b i P j=1,...,k (Tj \ S). Combining these two inequalities, we obtain ntumin() ktumin() TUmax(). REMARK 2. The final line of the proof of Theorem 2 shows that, in fat, the upper bound on TUmax()/TUmin() an be strengthened to the size of the winning set, k. Note that in Proposition 5, as well as in Corollaries 1 and 2, k = n 1, so these results do not ontradit eah other. For path autions, this upper bound an be improved to 2, mathing the lower bounds of Setion 4.1. THEOREM 3. For any instane of the shortest path problem, TUmax() 2 TUmin(). PROOF. Given a network (G, s, t), assume without loss of generality that the lexiographially-least heapest s t path, P, in G is {e 1,..., e k }, where e 1 = (s, v 1), e 2 = (v 1, v 2),..., e k = (v k 1, t). Let 1,..., k be the true osts of e 1,..., e k, and let b = (b 1,..., b k) and b = (b 1,..., b k) be bid vetors that orrespond to TUmin() and TUmax(), respetively. For any i = 1,..., k, there is a onstraint in (2) that is tight for b i with respet to the bid vetor b, i.e., an s t path P i that avoids e i and satisfies b (P \P i) = (P i\p). We an assume without loss of generality that P i oinides with P up to some vertex x i, then deviates from P to avoid e i, and finally returns to P at a vertex

y i and oinides with P from then on (learly, it might happen that s = x i or t = y i). Indeed, if P i deviates from P more than one, one of these deviations is not neessary to avoid e i and an be replaed with the respetive segment of P without inreasing the ost of P i. Among all paths of this form, let ˆP i be the one with the largest value of y i, i.e., the rightmost one. This path orresponds to an inequality I i of the form b x i +1 + + b y i ( ˆP i \ P). As in the proof of Theorem 2, we onstrut a set of tight onstraints L suh that every variable b i appears in at least one of these onstraints; however, now we have to be more areful about the hoie of onstraints in L. We onstrut L indutively as follows. Start by setting L = {I 1}. At the jth step, suppose that all variables up to (but not inluding) b i j appear in at least one inequality in L. Add I ij to L. Note that for any j we have y ij+1 > y ij. This is beause the inequalities added to L during the first j steps did not over b i j+1. See Figure 3. Sine y ij+2 > y ij+1, we must also have x ij+2 > y ij : otherwise, ˆPij+1 would not be the rightmost onstraint for b i j+1. Therefore, the variables in I ij+2 and I ij do not overlap, and hene no b i an appear in more than two inequalities in L. Now we follow the argument of the proof of Theorem 2 to finish. By adding up all of the (tight) inequalities in L for b i we obtain 2 P i=1,...,k b i P j=1,...,k ( ˆP j \ P). On the other hand, all these inequalities appear in ondition (2), so they must hold for b i, i.e., P i=1,...,k b i P j=1,...,k ( ˆP j \ P), so TUmax() 2TUmin(). 5. TRUTHFUL MECHANISMS FOR VER- TEX COVER Reall that for a vertex-over aution on a graph G = (V, E), an alloation rule is an algorithm that takes as input a bid b v for eah vertex and returns a vertex over Ŝ of G. As explained in Setion 2, we an ombine a monotone alloation rule with threshold payments to obtain a truthful aution. Two natural examples of monotone alloation rules are A opt, i.e., the algorithm that finds an optimal vertex over, and the greedy algorithm A GR. However, A opt annot be guaranteed to run in polynomial time unless P = NP and A GR has approximation ratio of log n. Another approximation algorithm for vertex over, whih has approximation ratio 2, is the loal ratio algorithm A LR [2, 3]. This algorithm onsiders the edges of G one by one. Given an edge e = (u, v), it omputes ǫ = min{b u, b v} and sets b u = b u ǫ, b v = b v ǫ. After all edges have been proessed, A LR returns the set of verties {v b v = 0}. It is not hard to hek that if the order in whih the edges are onsidered is independent of the bids, then this algorithm is monotone as well. Hene, we an use it to onstrut a truthful aution that is guaranteed to selet a vertex over whose ost is within a fator of 2 from the optimal. However, while the quality of the solution produed by A LR is muh better than that of A GR, we still need to show that its total payment is not too high. In the next subsetion, we bound the frugality ratio of A LR (and, more generally, all algorithms that satisfy the ondition of loal optimality, defined later) by 2, where is the maximum degree of G. We then prove a mathing lower bound showing that for some graphs the frugality ratio of any truthful aution is at least /2. 5.1 Upper bound P We say that an alloation rule is loally optimal if whenever b v > w v bw, the vertex v is not hosen. Note that for any suh rule the threshold bid of v satisfies t v P w v bw. CLAIM 1. The algorithms A opt, A GR, and A LR are loally optimal. THEOREM 4. Any vertex over aution M that has a loally optimal and monotone alloation rule and pays eah agent his threshold bid has frugality ratio φ NTUmin(M) 2. To prove Theorem 4, we first show that the total payment of any loally optimal mehanism does not exeed (V ). We then demonstrate that NTUmin() (V )/2. By ombining these two results, the theorem follows. LEMMA 5. Consider a graph G = (V, E) with maximum degree. Let M be a vertex-over aution on G that satisfies the onditions of Theorem 4. Then for any ost vetor, the total payment of M satisfies p M() (V ). PROOF. First note that any suh aution is truthful, so we an assume that eah agent s bid is equal to his ost. Let Ŝ be the vertex over seleted by M. Then by loal optimality p M() = X t v X X w X w = (V ). v Ŝ v Ŝ w v w V We now derive a lower bound on TUmax(); while not essential for the proof of Theorem 4, it helps us build the intuition neessary for that proof. LEMMA 6. For a vertex over instane G = (V, E) in whih S is a minimum vertex over, TUmax(, S) (V \ S) PROOF. For a vertex w with at least one neighbour in S, let d(w) denote the number of neighbours that w has in S. Consider the bid vetor b in whih, for eah v S, b v = P w w v,w S. d(w) Then P v S bv = P P v S w v,w S w/d(w) = P w/ S w = (V \ S). To finish we want to show that b is feasible in the sense that it satisfies (2). Consider a vertex over T, and extend the bid vetor b by assigning b v = v for v / S. Then b(t) = (T \S)+b(S T) (T \S)+ X X v S T w S T:w v w d(w), and sine all edges between S T and S go to S T, the righthand-side is equal to (T \S)+ X w = (T \S)+(S T) = (V \S) = b(s). w S T Next, we prove a lower bound on NTUmax(, S); we will then use it to obtain a lower bound on NTUmin(). LEMMA 7. For a vertex over instane G = (V, E) in whih S is a minimum vertex over, NTUmax(, S) (V \ S) PROOF. If (S) (V \ S), by ondition (1) we are done. Therefore, for the rest of the proof we assume that (S) < (V \ S). We show how to onstrut a bid vetor (b e) e S that satisfies onditions (1) and (2) suh that b(s) (V \ S); learly, this implies NTUmax(, S) (V \ S). Reall that a network flow problem is desribed by a direted graph Γ = (V Γ, E Γ), a soure node s V Γ, a sink node t V Γ, and a vetor of apaity onstraints a e, e E Γ. Consider a network (V Γ, E Γ) suh that V Γ = V {s, t}, E Γ = E 1 E 2 E 3, where E 1 = {(s, v) v S}, E 2 = {(v, w) v S, w

V \ S,(v, w) E}, E 3 = {(w, t) w V \ S}. Sine S is a vertex over for G, no edge of E an have both of its endpoints in V \ S, and by onstrution, E 2 ontains no edges with both endpoints in S. Therefore, the graph (V, E 2) is bipartite with parts (S, V \ S). Set the apaity onstraints for e E Γ as follows: a (s,v) = v, a (w,t) = w, a (v,w) = + for all v S, w V \ S. Reall that a ut is a partition of the verties in V Γ into two sets C 1 and C 2 so that s C 1, t C 2; we denote suh a ut by C = (C 1, C 2). Abusing notation, we write e = (u, v) C if u C 1, v C 2 or u C 2, v C 1, and say that suh an edge e = (u, v) rosses the ut C. The apaity of a ut C is omputed as ap(c) = P (v,w) C a (v,w). We have ap(s, V {t}) = (S), ap({s} V, t) = (V \ S). Let C min = ({s} S W, {t} S W ) be a minimum ut in Γ, where S, S S, W, W V \ S. See Figure 4. As ap(c min) ap(s, V {t}) = (S) < +, and any edge in E 2 has infinite apaity, no edge (u, v) E 2 rosses C min. Consider the network Γ = (V Γ, E Γ ), where V Γ = {s} S W {t}, E Γ = {(u, v) E Γ u, v V Γ }. Clearly, C = ({s} S W, {t}) is a minimum ut in Γ (otherwise, there would exist a smaller ut for Γ). As ap(c ) = (W ), we have (S ) (W ). Now, onsider the network Γ = (V Γ, E Γ ), where V Γ = {s} S W {t}, E Γ = {(u, v) E Γ u, v V Γ }. Similarly, C = ({s}, S W {t}) is a minimum ut in Γ, ap(c ) = (S ). As the size of a maximum flow from s to t is equal to the apaity of a minimum ut separating s and t, there exists a flow F = (f e) e EΓ of size (S ). This flow has to saturate all edges between s and S, i.e., f (s,v) = v for all v S. Now, inrease the apaities of all edges between s and S to +. In the modified network, the apaity of a minimum ut (and hene the size of a maximum flow) is (W ), and a maximum flow F = (f e) e EΓ an be onstruted by greedily augmenting F. Set b v = v for all v S, b v = f (s,v) for all v S. As F is onstruted by augmenting F, we have b v v for all v S, i.e., ondition (1) is satisfied. Now, let us hek that no vertex over T V an violate ondition (2). Set T 1 = T S, T 2 = T S, T 3 = T W, T 4 = T W ; our goal is to show that b(s \ T 1) +b(s \ T 2) (T 3)+(T 4). Consider all edges (u, v) E suh that u S \T 1. If (u, v) E 2 then v T 3 (no edge in E 2 an ross the ut), and if u, v S then v T 1 T 2. Hene, T 1 T 3 S is a vertex over for G, and therefore (T 1)+(T 3)+(S ) (S) = (T 1)+(S \ T 1) + (S ). Consequently, (T 3) (S \ T 1) = b(s \ T 1). Now, onsider the verties in S \T 2. Any edge in E 2 that starts in one of these verties has to end in T 4 (this edge has to be overed by T, and it annot go aross the ut). Therefore, the total flow out of S \T 2 is at most the total flow out of T 4, i.e., b(s \T 2) (T 4). Hene, b(s \ T 1) + b(s \ T 2) (T 3) + (T 4). Finally, we derive a lower bound on the payment bound that is of interest to us, namely, NTUmin(). LEMMA 8. For a vertex over instane G = (V, E) in whih S is a minimum vertex over, NTUmin(, S) (V \ S) PROOF. Suppose for ontradition that is a ost vetor with minimum-ost vertex over S and NTUmin(, S) < (V \S). Let b be the orresponding bid vetor and let be a new ost vetor with v = b v for v S and v = v for v S. Condition (2) guarantees that S is an optimal solution to the ost vetor. Now ompute a bid vetor b orresponding to NTUmax(, S). We W S s 0 1 0 1 1 T 2 T 1 00 00 01 01 00 0 100 T 3 T 4 0 0 1 1 00 00 0 1 1 00 00 t S W Figure 4: Proof of Lemma 7. Dashed lines orrespond to edges in E \ E 2 laim that b v = v for any v S. Indeed, suppose that b v > v for some v S (b v = v for v S by onstrution). As b satisfies onditions (1) (3), among the inequalities in (2) there is one that is tight for v and the bid vetor b. That is, b(s \ T) = (T \ S). By the onstrution of, (S \ T) = (T \ S). Now sine b w w for all w S, b v > v implies b (S \T) > (S \T) = (T \S). But this violates (2). So we now know b =. Hene, we have NTUmax(, S) = P v S bv = NTUmin(, S) < (V \ S), giving a ontradition to the fat that NTUmax(, S) (V \S) whih we proved in Lemma 7. As NTUmin(, S) satisfies ondition (1), it follows that we have NTUmin(, S) (S). Together will Lemma 8, this implies NTUmin(, S) max{(v \ S), (S)} (V )/2. Combined with Lemma 5, this ompletes the proof of Theorem 4. REMARK 3. As NTUmin() NTUmax() TUmax(), our bound of 2 extends to the smaller frugality ratios that we onsider, i.e., φ NTUmax(M) and φ TUmax(M). It is not lear whether it extends to the larger frugality ratio φ TUmin(M). However, the frugality ratio φ TUmin(M) is not realisti beause the payment bound TUmin() is inappropriately low we show in Setion 6 that TUmin() an be signifiantly smaller than the total ost of a heapest vertex over. Extensions We an also apply our results to monotone vertex-over algorithms that do not neessarily output loally-optimal solutions. To do so, we simply take the vertex over produed by any suh algorithm and transform it into a loally-optimal one, onsidering the verties in lexiographi order and replaing a vertex v with its neighbours whenever b v > P u v bu. Note that if a vertex u has been added to the vertex over during this proess, it means that it has a neighbour whose bid is higher than b u, so after one pass all verties in the vertex over satisfy b v P u v bu. This proedure is monotone in bids, and it an only derease the ost of the vertex over. Therefore, using it on top of a monotone alloation rule with approx-

imation ratio α, we obtain a monotone loally-optimal alloation rule with approximation ratio α. Combining it with threshold payments, we get an aution with φ NTUmin 2. Sine any truthful aution has a monotone alloation rule, this proedure transforms any truthful mehanism for the vertex-over problem into a frugal one while preserving the approximation ratio. 5.2 Lower bound In this subsetion, we prove that the upper bound of Theorem 4 is essentially optimal. Our proof uses the tehniques of [9], where the authors prove a similar result for shortest-path autions. THEOREM 5. For any > 0 and any n, there exist a graph G of maximum degree and size N > n suh that for any truthful mehanism M on G we have φ NTUmin(M) /2. PROOF. Given n and, set k = n/2. Let G be the graph that onsists of k bloks B 1,..., B k of size 2 eah, where eah B i is a omplete bipartite graph with parts L i and R i, L i = R i =. We will onsider two families of ost vetors for G. Under a ost vetor x X, eah blok B i has one vertex of ost 1; all other verties ost 0. Under a ost vetor y Y, there is one blok that has two verties of ost 1, one in eah part, all other bloks have one vertex of ost 1, and all other verties ost 0. Clearly, X = (2 ) k, Y = k(2 ) k 1 2. We will now onstrut a bipartite graph W with the vertex set X Y as follows. Consider a ost vetor y Y that has two verties of ost 1 in B i; let these verties be v l L i and v r R i. By hanging the ost of either of these verties to 0, we obtain a ost vetor in X. Let x l and x r be the ost vetors obtained by hanging the ost of v l and v r, respetively. The vertex over hosen by M(y) must either ontain all verties in L i or it must ontain all verties in R i. In the former ase, we put in W an edge from y to x l and in the latter ase we put in W an edge from y to x r (if the vertex over inludes all of B i, W ontains both of these edges). The graph W has at least k(2 ) k 1 2 edges, so there must exist an x X of degree at least k /2. Let y 1,..., y k /2 be the other endpoints of the edges inident to x, and for eah i = 1,..., k /2, let v i be the vertex whose ost is different under x and y i; note that all v i are distint. It is not hard to see that NTUmin(x) k: the heapest vertex over ontains the all-0 part of eah blok, and we an satisfy onditions (1) (3) by letting one of the verties in the all-0 part of eah blok to bid 1, while all other the verties in the heapest set bid 0. On the other hand, by monotoniity of M we have v i M(x) for i = 1,..., k /2 (v i is in the winning set under y i, and x is obtained from y i by dereasing the ost of v i), and moreover, the threshold bid of eah v i is at least 1, so the total payment of M onx is at least k /2. Hene, φ NTUmin(M) M(x)/NTUmin(x) /2. REMARK 4. The lower bound of Theorem 5 an be generalised to randomised mehanisms, where a randomised mehanism is onsidered to be truthful if it an be represented as a probability distribution over truthful mehanisms. In this ase, instead of hoosing the vertex x X with the highest degree, we put both (y,x l ) and (y,x r) into W, label eah edge with the probability that the respetive part of the blok is hosen, and pik x X with the highest weighted degree. The argument an be further extended to a more permissive definition of truthfulness for randomised mehanisms, but this disussion is beyond the sope of this paper. 6. PROPERTIES OF PAYMENT BOUNDS In this setion we onsider several desirable properties of payment bounds and evaluate the four payment bounds proposed in this paper with respet to them. The partiular properties that we are interested in are independene of the hoie of S (Setion 6.3), monotoniity (Setion 6.4.1), omputational hardness (Setion 6.4.2), and the relationship with other reasonable bounds, suh as the total ost of the heapest set (Setion 6.1), or the total VCG payment (Setion 6.2). 6.1 Comparison with total ost Our first requirement is that a payment bound should not be less than the total ost of the seleted set. Payment bounds are used to evaluate the performane of set-system autions. The latter have to satisfy individual rationality, i.e., the payment to eah agent must be at least as large as his inurred osts; it is only reasonable to require the payment bound to satisfy the same requirement. Clearly, NTUmax() and NTUmin() satisfy this requirement due to ondition (1), and so does TUmax(), sine TUmax() NTUmax(). However, TUmin() fails this test. The example of Proposition 4 shows that for path autions, TUmin() an be smaller than the total ost by a fator of 2. Moreover, there are set systems and ost vetors for whih TUmin() is smaller than the ost of the heapest set S by a fator of Ω(n). Consider, for example, the vertex-over aution for the graph of Proposition 5 with the osts X1 = = Xn 2 = Xn 1 = 1, X0 = 0. The ost of a heapest vertex over is n 2, and the lexiographially first vertex over of ost n 2 is {X 0, X 1,..., X n 2}. The onstraints in (2) are b Xi + b X0 Xn 1 = 1. Clearly, we an satisfy onditions (2) and (3) by setting b X1 = = b Xn 2 = 0, b X0 = 1, whih means that TUmin() 1. This observation suggests that the payment bound TUmin() is too strong to be realisti, sine it an be substantially lower than the ost of the heapest feasible set. Nevertheless, some of the positive results that were proved in [16] for NTUmin() go through for TUmin() as well. In partiular, one an show that if the feasible sets are the bases of a monopolyfree matroid, then φ TUmin(VCG) = 1. To show that φ TUmin(VCG) is at most 1, one must prove that the VCG payment is at most TUmin(). This is shown for NTUmin() in the first paragraph of the proof of Theorem 5 in [16]. Their argument does not use ondition (1) at all, so it also applies to TUmin(). On the other hand, φ TUmin(VCG) 1 sine φ TUmin(VCG) φ NTUmin(VCG) and φ NTUmin(VCG) 1 by Proposition 7 of [16] (and also by Proposition 6 below). 6.2 Comparison with VCG payments Another measure of suitability for payment bounds is that they should not result in frugality ratios that are less then 1 for wellknown truthful mehanisms. If this is indeed the ase, the payment bound may be too weak, as it beomes too easy to design mehanisms that perform well with respet to it. It partiular, a reasonable requirement is that a payment bound should not exeed the total payment of the lassial VCG mehanism. The following proposition shows that NTUmax(), and therefore also NTUmin() and TUmin(), do not exeed the VCG payment p VCG(). The proof essentially follows the argument of Proposition 7 of [16] and an be found in the full version of this paper [8]. PROPOSITION 6. φ NTUmax(VCG) 1. Proposition 6 shows that none of the payment bounds TUmin(), NTUmin() and NTUmax() exeeds the payment of VCG. However, the payment bound TUmax() an be larger that the total

VCG payment. In partiular, for the instane in Proposition 5, the VCG payment is smaller than TUmax() by a fator of n 2. We have already seen that TUmax() n 2. On the other hand, under VCG, the threshold bid of any X i, i = 1,..., n 2, is 0: if any suh vertex bids above 0, it is deleted from the winning set together with X 0 and replaed with X n 1. Similarly, the threshold bid of X 0 is 1, beause if X 0 bids above 1, it an be replaed with X n 1. So the VCG payment is 1. This result is not surprising: the definition of TUmax() impliitly assumes there is o-operation between the agents, while the omputation of VCG payments does not take into aount any interation between them. Indeed, o-operation enables the agents to extrat higher payments under VCG. That is, VCG is not groupstrategyproof. This suggests that as a payment bound, TUmax() may be too liberal, at least in a ontext where there is little or no o-operation between agents. Perhaps TUmax() an be a good benhmark for measuring the performane of mehanisms designed for agents that an form oalitions or make side payments to eah other, in partiular, group-strategyproof mehanisms. Another setting in whih bounding φ TUmax is still of some interest is when, for the underlying problem, the optimal alloation and VCG payments are NP-hard to ompute. In this ase, finding a polynomial-time omputable mehanism with good frugality ratio with respet to TUmax() is a non-trivial task, while bounding the frugality ratio with respet to more hallenging payment bounds ould be too diffiult. To illustrate this point, ompare the proofs of Lemma 6 and Lemma 7: both require some effort, but the latter is muh more diffiult than the former. 6.3 The hoie of S All payment bounds defined in this paper orrespond to the total bid of all elements in the heapest feasible set, where ties are broken lexiographially. While this definition ensures that our payment bounds are well-defined, the partiular hoie of the drawresolution rule appears arbitrary, and one might wonder if our payment bounds are suffiiently robust to be independent of this hoie. It turns out that is indeed the ase for NTUmin() and NTUmax(), i.e., these bounds do not depend on the draw-resolution rule. To see this, suppose that two feasible sets S 1 and S 2 have the same ost. In the omputation of NTUmin(, S 1), all verties in S 1 \S 2 would have to bid their true ost, sine otherwise S 2 would beome heaper than S 1. Hene, any bid vetor for S 1 an only have b e e for e S 1 S 2, and hene onstitutes a valid bid vetor for S 2 and vie versa. A similar argument applies to NTUmax(). However, for TUmin() and TUmax() this is not the ase. For example, onsider the set system E = {e 1, e 2, e 3, e 4, e 5}, F = {S 1 = {e 1, e 2}, S 2 = {e 2, e 3, e 4}, S 3 = {e 4, e 5}} with the osts 1 = 2, 2 = 3 = 4 = 1, 5 = 3. The heapest sets are S 1 and S 2. Now TUmax(, S 1) 4, as the total bid of the elements in S 1 annot exeed the total ost of S 3. On the other hand, TUmax(, S 2) 5, as we an set b 2 = 3, b 3 = 0, b 4 = 2. Similarly, TUmin(, S 1) = 4, beause the inequalities in (2) are b 1 2 and b 1 + b 2 4. But TUmin(, S 2) 3 as we an set b 2 = 1, b 3 = 2, b 4 = 0. 6.4 Negative results for NTUmin() and TUmin() The results in [16] and our vertex over results are proved for the frugality ratio φ NTUmin. Indeed, it an be argued that φ NTUmin is the best definition of frugality ratio, beause among all reasonable payment bounds (i.e., ones that are at least as large as the ost of the heapest feasible set), it is most demanding of the algorithm. However, NTUmin() is not always the easiest or the most natural payment bound to work with. In this subsetion, we disuss several disadvantages of NTUmin() (and also TUmin()) as ompared to NTUmax() and TUmax(). 6.4.1 Nonmonotoniity The first problem with NTUmin() is that it is not monotone with respet to F, i.e., it may inrease when one adds a feasible set to F. (It is, however, monotone in the sense that a losing agent annot beome a winner by raising his ost.) Intuitively, a good payment bound should satisfy this monotoniity requirement, as adding a feasible set inreases the ompetition, so it should drive the pries down. Note that this indeed the ase for NTUmax() and TUmax() sine a new feasible set adds a onstraint in (2), thus limiting the solution spae for the respetive linear program. PROPOSITION 7. Adding a feasible set to F an inrease the value of NTUmin() by a fator of Ω(n). PROOF. Let E = {x, xx,y 1,..., y n, z 1,..., z n}. Set Y = {y 1,..., y n}, S = Y {x}, T i = Y \ {y i} {z i}, i = 1,..., n, and suppose that F = {S, T 1,..., T n}. The osts are x = 0, xx = 0, yi = 0, zi = 1 for i = 1,..., n. Note that S is the heapest feasible set. Let F = F {T 0}, where T 0 = Y {xx}. For F, the bid vetor b y1 = = b yn = 0, b x = 1 satisfies (1), (2), and (3), so NTUmin() 1. For F, S is still the lexiographially-least heapest set. Any optimal solution has b x = 0 (by onstraint in (2) with T 0). Condition (3) for y i implies b x + b yi = zi = 1, so b yi = 1 and NTUmin() = n. For path autions, it has been shown [18] that NTUmin() is non-monotone in a slightly different sense, i.e., with respet to adding a new edge (agent) rather than a new feasible set (a team of existing agents). REMARK 5. We an also show that NTUmin() is non-monotone for vertex over. In this ase, adding a new feasible set orresponds to deleting edges from the graph. It turns out that deleting a single edge an inrease NTUmin() by a fator of n 2; the onstrution is similar to that of Proposition 5. 6.4.2 NP-Hardness Another problem with NTUmin(, S) is that it is NP-hard to ompute even if the number of feasible sets is polynomial in n. Again, this puts it at a disadvantage ompared to NTUmax(, S) and TUmax(, S) (see Remark 1). THEOREM 6. Computing NTUmin() is NP-hard, even when the lexiographially-least feasible set S is given in the input. PROOF. We redue EXACT COVER BY 3-SETS(X3C) to our problem. An instane of X3C is given by a universe G = {g 1,..., g n} and a olletion of subsets C 1,..., C m, C i G, C i = 3, where the goal is to deide whether one an over G by n/3 of these sets. Observe that if this is indeed the ase, eah element of G is ontained in exatly one set of the over. LEMMA 9. Consider a minimisation problem P of the following form: Minimise P i=1,...,n bi under onditions (1) bi 0 for all i = 1,..., n; (2) for any j = 1,..., k we have P b i S j b i a j, where S j {b 1,..., b n}; (3) for eah b j, one of the onstraints in (2) involving it is tight. For any suh P, one an onstrut a set system S and a vetor of osts suh that NTUmin() is the optimal solution to P. PROOF. The onstrution is straightforward: there is an element of ost 0 for eah b i, an element of ost a j for eah a j, the feasible solutions are {b 1,..., b n}, or any set obtained from {b 1,..., b n} by replaing the elements in S j by a j.