Combinatorial Auctions: VC v. VCG Elchanan Mossel ∗ Christos Papadimitriou † Michael Schapira ‡ Yaron Singer § Abstract The existence of incentive-compatible, computationally-efficient protocols for combinatorial auctions with decent approximation ratios is one of the most central and well studied open questions in mechanism design. The only universal technique known for the design of truth- ful mechanisms is the celebrated Vickrey-Clarke-Groves (VCG) scheme, which is “maximal in range ”, i.e., it always exactly optimizes over a subset of the possible outcomes. We present a first-of-its-kind technique for proving computational-complexity inapproximability results for maximal-in-range mechanism for combinatorial auctions (under the complexity assumption that NP has no polynomial circuits). We show that in some interesting cases the lower bounds ob- tained using this technique can be extended to hold for all truthful mechanisms. Our lower- bounding method is based on a generalization of the VC-dimension to k-tuples of disjoint sets. We illustrate our technique via the case of two-bidder combinatorial auctions. We believe that this technique is of independent interest, and has great promise for making progress on the general problem. ∗ Statistics and Computer Science, U.C. Berkeley, and Mathematics and Computer Science Weizmann Institute. Supported by Sloan fellowship in Mathematics, NSF Career award DMS 0548249, DOD grant N0014-07-1-05-06, and by ISF. mossel@stat.berkeley.edu † Computer Science Division University of California at Berkeley, CA, 94720 USA. christos@cs.berkeley.edu ‡ Department of Computer Science, Yale University, CT, USA, and Computer Science Division, University of California at Berkeley, CA, USA. Supported by NSF grant 0331548. michael.schapira@yale.edu. § Computer Science Division University of California at Berkeley, CA, 94720 USA. Supported by grants XXXXX. yaron@cs.berkeley.edu. 0