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.
 We consider computationally-efficient incentive-compatiblemechanisms that use the VCG payment scheme, and study how well theycan approximate the social welfare in auction settings. We present anovel technique for setting lower bounds on the approximation ratioof this type of mechanisms. Specifically, for combinatorial auctionsamong submodular (and thus also subadditive) bidders we prove an Ω(m1 6) lower bound, which is close to the knownupper bound of O(m1 2), and qualitatively higher than theconstant factor approximation possible from a purely computationalpoint of view. Let M e 0, 1, 2, ..., m —1 , N e 0, 1, 2,..., n —1 , and f:M × N → 0, 1 a Boolean-valued function. We will be interested in the following problem and its related questions. Let i e M , j e N be integers known only to two persons P 1 and P 2 , respectively. For P 1 and P 2 to determine cooperatively the value f ( i, j ), they send information to each other alternately, one bit at a time, according to some algorithm. The quantity of interest, which measures the information exchange necessary for computing f , is the minimum number of bits exchanged in any algorithm. For example, if f ( i, j ) e ( i + j ) mod 2. then 1 bit of information (conveying whether i is odd) sent from P 1 to P 2 will enable P 2 to determine f ( i, j ), and this is clearly the best possible. The above problem is a variation of a model of Abelson [1] concerning information transfer in distributive computions.