The existence of incentive-compatible computationally-efficient protocols for combinatorial auctions with decent approximation ratios is the paradigmatic problem in computational mechanism design. It is believed that in many cases good approximations for combinatorial auctions may be unattainable due to an inherent clash between truthfulness and computational efficiency. However, to date, researchers lack the machinery to prove such results. In this paper, we present a new approach that we believe holds great promise for making progress on this important problem. We take the first steps towards the development of new technologies for lower bounding the VC-dimension of k-tuples of disjoint sets. We apply this machinery to prove the first computational-complexity inapproximability results for incentive-compatible mechanisms for combinatorial auctions. These results hold for the important class of VCG-based mechanisms, and are based on the complexity assumption that NP has no polynomial-size circuits.
Combinatorial auctions have been extensively studied in both the economics and the computer science literature @cite_44 @cite_18 @cite_34 . It is known that if the preferences of the bidders are unrestricted then no constant approximation ratios are achievable (in polynomial time) @cite_36 @cite_46 . Hence, much research has been devoted to the exploration of restrictions on bidders' preferences that allow for good approximations, , for , and , preferences approximation ratios have been obtained @cite_16 @cite_2 @cite_12 @cite_35 @cite_15 @cite_33 . In contrast, the known approximation algorithms for these classes have approximation ratios @cite_29 @cite_16 @cite_5 . It is believed that this gap may be due to the computational burden imposed by the truthfulness requirement. However, to date, this belief remains unproven. In particular, no lower bounds for truthful mechanisms for combinatorial auctions are known.
@cite_35 @cite_18 @cite_33 @cite_15 @cite_36 @cite_29 @cite_16 @cite_44 @cite_2 @cite_5 @cite_46 @cite_34 @cite_12
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Combinatorial allocation problems require allocating items to players in a way that maximizes the total utility. Two such problems received attention recently, and were addressed using the same linear programming (LP) relaxation. In the Maximum Submodular Welfare (SMW) problem, utility functions of players are submodular, and for this case Dobzinski and Schapira [SODA 2006] showed an approximation ratio of 1 - 1 e. In the Generalized Assignment Problem (GAP) utility functions are linear but players also have capacity constraints. GAP admits a (1 - 1 e)- approximation as well, as shown by Fleischer, Goemans, Mirrokni and Sviridenko [SODA 2006]. In both cases, the approximation ratio was in fact shown for a more general version of the problem, for which improving 1 - 1 e is NPhard. In this paper, we show how to improve the 1 - 1 e approximation ratio, both for SMW and for GAP. A common theme in both improvements is the use of a new and optimal Fair Contention Resolution technique. However, each of the improvements involves a different rounding procedure for the above mentioned LP. In addition, we prove APX-hardness results for SMW (such results were known for GAP). An important feature of our hardness results is that they apply even in very restricted settings, e.g. when every player has nonzero utility only for a constant number of items. Many auctions involve the sale of a variety of distinct assets. Examples are airport time slots, delivery routes, network routing, and furniture. Because of complementarities or substitution effects between the different assets, bidders have preferences not just for particular items but for sets of items. For this reason, economic efficiency is enhanced if bidders are allowed to bid on bundles or combinations of different assets. This paper surveys the state of knowledge about the design of combinatorial auctions and presents some new insights. Periodic updates of portions of this survey will be posted to this journal's Online Supplements web page at http: joc.pubs.informs.org OnlineSupplements.html. In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2[m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility ∑i=1n wi(Si). In this paper, we work in the value oracle model where the only access to the utility functions is through a black box returning wi(S) for a given set S. Submodular Welfare is in fact a special case of the more general problem of submodular maximization subject to a matroid constraint: max f(S): S ∈ I , where f is monotone submodular and I is the collection of independent sets in some matroid. For both problems, a greedy algorithm is known to yield a 1 2-approximation [21, 16]. In special cases where the matroid is uniform (I = S: |S| ≤ k) [20] or the submodular function is of a special type [4, 2], a (1-1 e)-approximation has been achieved and this is optimal for these problems in the value oracle model [22, 6, 15]. A (1-1 e)-approximation for the general Submodular Welfare Problem has been known only in a stronger demand oracle model [4], where in fact 1-1 e can be improved [9]. In this paper, we develop a randomized continuous greedy algorithm which achieves a (1-1 e)-approximation for the Submodular Welfare Problem in the value oracle model. We also show that the special case of n equal players is approximation resistant, in the sense that the optimal (1-1 e)-approximation is achieved by a uniformly random solution. Using the pipage rounding technique [1, 2], we obtain a (1-1 e)-approximation for submodular maximization subject to any matroid constraint. The continuous greedy algorithm has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and the AdWords Assignment Problem. Some important classical mechanisms considered in Microeconomics and Game Theory require the solution of a difficult optimization problem. This is true of mechanisms for combinatorial auctions, which have in recent years assumed practical importance, and in particular of the gold standard for combinatorial auctions, the Generalized Vickrey Auction (GVA). Traditional analysis of these mechanisms - in particular, their truth revelation properties - assumes that the optimization problems are solved precisely. In reality, these optimization problems can usually be solved only in an approximate fashion. We investigate the impact on such mechanisms of replacing exact solutions by approximate ones. Specifically, we look at a particular greedy optimization method, which has empirically been shown to perform well. We show that the GVA payment scheme does not provide for a truth revealing mechanism. We introduce another scheme that does guarantee truthfulness for a restricted class of players. We demonstrate the latter property by identifying sufficient conditions for a combinatorial auction to be truth-revealing, conditions which have applicability beyond the specific auction studied here. This paper discusses two advancements in the theory of designing truthful randomized mechanisms. Our first contribution is a new framework for developing truthful randomized mechanisms. The framework enables the construction of mechanisms with polynomially small failure probability. This is in contrast to previous mechanisms that fail with constant probability. Another appealing feature of the new framework is that bidding truthfully is a stronglydominant strategy. The power of the framework is demonstrated by an @math -mechanism for combinatorial auctions that succeeds with probability @math . The other major result of this paper is an O(logmloglogm) randomized truthful mechanism for combinatorial auction with subadditivebidders. The best previously-known truthful mechanism for this setting guaranteed an approximation ratio of @math . En route, the new mechanism also provides the best approximation ratio for combinatorial auctions with submodularbidders currently achieved by truthful mechanisms. We exhibit three approximation algorithms for the allocation problem in combinatorial auctions with complement free bidders. The running time of these algorithms is polynomial in the number of items @math and in the number of bidders n, even though the "input size" is exponential in m. The first algorithm provides an O(log m) approximation. The second algorithm provides an O(√ m) approximation in the weaker model of value oracles. This algorithm is also incentive compatible. The third algorithm provides an improved 2-approximation for the more restricted case of "XOS bidders", a class which strictly contains submodular bidders. We also prove lower bounds on the possible approximations achievable for these classes of bidders. These bounds are not tight and we leave the gaps as open problems. We explore the allocation problem in combinatorial auctions with submodular bidders. We provide an e e-1 approximation algorithm for this problem. Moreover, our algorithm applies to the more general class of XOS bidders. By presenting a matching unconditional lower bound in the communication model, we prove that the upper bound is tight for the XOS class.Our algorithm improves upon the previously known 2-approximation algorithm. In fact, we also exhibit another algorithm which obtains an approximation ratio better than 2 for submodular bidders, even in the value queries model.Throughout the paper we highlight interesting connections between combinatorial auctions with XOS and submodular bidders and various other combinatorial optimization problems. In particular, we discuss coverage problems and online problems. We present a new framework for the design of computationally-efficient and incentive-compatible mechanisms for combinatorial auctions. The mechanisms obtained via this framework are randomized, and obtain incentive compatibility in the universal sense (in contrast to the substantially weaker notion of incentive compatibility in expectation). We demonstrate the usefulness of our techniques by exhibiting two mechanisms for combinatorial auctions with general bidder preferences. The first mechanism obtains an optimal O(m)-approximation to the optimal social welfare for arbitrary bidder valuations. The second mechanism obtains an O(log^2m)-approximation for a class of bidder valuations that contains the important class of submodular bidders. These approximation ratios greatly improve over the best (known) deterministic incentive-compatible mechanisms for these classes. We show that any communication finding a value-maximizing allocation in a private-information economy must also discover supporting prices (in general personalized and nonlinear). In particular, to allocate L indivisible items between two agents, a price must be revealed for each of the 2L-1 bundles. We prove that all monotonic prices for an agent must be used, hence exponential communication in L is needed. Furthermore, exponential communication is needed just to ensure a higher share of surplus than that realized by auctioning all items as a bundle, or even a higher expected surplus (for some probability distribution over valuations). When the utilities are submodular, efficiency still requires exponential communication (and fully polynomial approximation is impossible). When the items are identical, arbitrarily good approximation is obtained with exponentially less communication than exact efficiency. We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions. Our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least half that of the value of the fractional solution. This approximation ratio of 1 2 improves over an Ω(1 log m) ratio of Dobzinski, Nisan and Schapira [STOC 2005]. We also show an approximation ratio of 1 - 1 e when utility functions are fractionally subadditive. A result similar to this last result was previously obtained by Dobzinski and Schapira [Soda 2006], but via a different rounding technique that requires the use of a so called "XOS oracle".The randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation, but have no access to the actual utility functions of the players. This allows us to suggest new incentive compatible mechanisms for combinatorial auctions, extending previous work of Lavi and Swamy [FOCS 2005].