Inapproximability for VCG-based combinatorial auctions

Authors:
Dave Buchfuhrer;Shaddin Dughmi;Hu Fu;Robert Kleinberg;Elchanan Mossel;Christos Papadimitriou;Michael Schapira;Yaron Singer;Chris Umans
Affiliations:
Caltech, Pasadena, CA;Stanford University;Cornell University;Cornell University, Ithaca, NY;U.C. Berkeley, and Mathematics and Computer Science Weizmann Institute;University of California at Berkeley, CA;Yale University, CT, and University of California at Berkeley, CA;University of California at Berkeley, CA;Caltech, Pasadena, CA
Venue:
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Year:
2010

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Abstract

The existence of incentive-compatible, computationally-efficient mechanisms for combinatorial auctions with good approximation ratios is the paradigmatic problem in algorithmic 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. In this paper, we prove the first computational-complexity in-approximability results for incentive-compatible mechanisms for combinatorial auctions. Our results are tight, hold for the important class of VCG-based mechanisms, and are based on the complexity assumption that NP has no polynomial-size circuits. We show two different techniques to obtain such lower bounds: one for deterministic mechanisms that attains optimal dependence on the number of players and number of items, and one that also applies to a class of randomized mechanisms and attains optimal dependence on the number of players. Both techniques are based on novel VC dimension machinery.