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
2950708966 2078273056 2333413293 2951660602
One of the fundamental questions of Algorithmic Mechanism Design is whether there exists an inherent clash between truthfulness and computational tractability: in particular, whether polynomial-time truthful mechanisms for combinatorial auctions are provably weaker in terms of approximation ratio than non-truthful ones. This question was very recently answered for universally truthful mechanisms for combinatorial auctions D11 , and even for truthful-in-expectation mechanisms DughmiV11 . However, both of these results are based on information-theoretic arguments for valuations given by a value oracle, and leave open the possibility of polynomial-time truthful mechanisms for succinctly described classes of valuations. This paper is the first to prove computational hardness results for truthful mechanisms for combinatorial auctions with succinctly described valuations. We prove that there is a class of succinctly represented submodular valuations for which no deterministic truthful mechanism provides an @math -approximation for a constant @math , unless @math ( @math denotes the number of items). Furthermore, we prove that even truthful-in-expectation mechanisms cannot approximate combinatorial auctions with certain succinctly described submodular valuations better than within @math , where @math is the number of bidders and @math some absolute constant, unless @math . In addition, we prove computational hardness results for two related problems. We show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that provides an approximation ratio of m1 2 -e must use exponentially many value queries, where m is the number of items. In contrast, ignoring incentives there exist constant ratio approximation algorithms for this problem. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far. We demonstrate two additional applications of our new technique: (1) an impossibility result for universally-truthful polynomial time flexible combinatorial public projects and (2) an impossibility result for truthful-in-expectation mechanisms for exact combinatorial public projects. The latter is the first result that bounds the power of polynomial-time truthful in expectation mechanisms in any setting. A long-standing open question in algorithmic mechanism design is whether there exist computationally efficient truthful mechanisms for combinatorial auctions, with performance guarantees close to those possible without considerations of truthfulness. In this article, we answer this question negatively: the requirement of truthfulness can impact dramatically the ability of a mechanism to achieve a good approximation ratio for combinatorial auctions. More precisely, we show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that approximates optimal social welfare within a factor of m1 2−e must use exponentially many value queries, where m is the number of items. Furthermore, we show that there exists a class of succinctly represented submodular valuation functions, for which the existence of a universally truthful polynomial-time mechanism that provides an m1 2−e-approximation would imply NP e RP. In contrast, ignoring truthfulness, there exist constant-factor approximation algorithms for this problem, and ignoring computational efficiency, the VCG mechanism is truthful and provides optimal social welfare. These are the first hardness results for truthful polynomial-time mechanisms for any type of combinatorial auctions, even for deterministic mechanisms. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far. We study a central problem in Algorithmic Mechanism Design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a non-trivial approximation ratio of @math [STOC'06], where @math is the number of items. This was subsequently improved to @math [Dobzinski, APPROX'07] and then to @math [Krysta and Vocking, ICALP'12]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of @math . Similarly to previous constructions, our mechanism uses polynomially many value and demand queries, and in fact provides the same approximation ratio for the larger class of XOS (a.k.a. fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although in general computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive.