A The Computational Complexity of Truthfulness in Combinatorial Auctions

Is it possible to design truthful polynomial-time mechanisms that achieve a good approximation ratio? This is one of the basic questions in Algorithmic Mechanism Design. The focus of most of the work on this question is on settings in which on one hand there exists a truthful algorithm that maximizes the social welfare (VCG), but on the other hand computing the optimal solution is computationally intractable. The goal is therefore to determine whether there exist truthful polynomial time mechanisms that guarantee good approximation ratios. Several settings were considered in the literature, but the flagship challenge is to design polynomial-time truthful approximation mechanisms for the problem of combinatorial auctions. In a combinatorial auction, we want to sell m items to n bidders with valuation functions vi : 2 → R+. As usual, we assume that vi(∅) = 0 and that vi is monotone, i.e. vi(S) ≤ vi(T ) whenever S ⊂ T . The goal is to design a mechanism that allocates disjoint sets S1, . . . , Sn to the n bidders, optimizing (at least approximately) the social welfare ∑n i=1 vi(Si), in a way that incentivizes the bidders to report their true valuations (the property of incentive-compatibility, or truthfulness). This is done by charging payments p1, . . . , pn by the mechanism, so that for each player, reporting his true valuation maximizes the profit vi(Si) − pi (in the case of truthfulness in expectation, the reporting the true valuation maximizes the expectation of this expression). Without the requirement of truthfulness, combinatorial auctions admit constant-factor approximation algorithms for various non-trivial classes of valuations functions, in particular for submodular valuations: A valuation v is called submodular if for each S and T we have that v(S) + v(T ) ≥ v(S ∩ T ) + v(S ∪ T ). Combinatorial auctions with submodular valuations admit a (1 − 1/e)-approximation [Vondrák 2008] and it is also known that this approximation is optimal [Khot et al. 2005]. On the other hand, the VCG mechanism is truthful and provides optimal social welfare, but naturally is not computationally efficient. Therefore, submodular valuations form a natural setting to investigate the question whether truthfulness can be reconciled with polynomial-time approximation or not. The best known polynomial time truthful mechanism for combinatorial auctions with submodular valuations achieves a factor of O( √ m) [Dobzinski et al. 2005]. This is quite a