text: We design an expected polynomial-time, truthful-in-expectation, (1-1 e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are m matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions, matroid weighted-rank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P != NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1 e)-approximation of the optimal welfare.
  @cite_15: This short note exhibits a truthful-in-expectation @math -approximation mechanism for combinatorial auctions with subadditive bidders that uses polynomial communication.
  @cite_18: This paper discusses two advancements in the theory of designing truthful randomized mechanisms. Our first contribution is a new framework for developing truthful randomized mechanisms. The framework enables the construction of mechanisms with polynomially small failure probability. This is in contrast to previous mechanisms that fail with constant probability. Another appealing feature of the new framework is that bidding truthfully is a stronglydominant strategy. The power of the framework is demonstrated by an @math -mechanism for combinatorial auctions that succeeds with probability @math . The other major result of this paper is an O(logmloglogm) randomized truthful mechanism for combinatorial auction with subadditivebidders. The best previously-known truthful mechanism for this setting guaranteed an approximation ratio of @math . En route, the new mechanism also provides the best approximation ratio for combinatorial auctions with submodularbidders currently achieved by truthful mechanisms.
summary: Despite intense study, prior to this work, there were no truthful-in-expectation and polynomial-time constant-factor approximation mechanisms for welfare maximization with any non-trivial subclass of submodular bidder valuations. The best previous results, which apply to all submodular valuations, are a truthful-in-expectation @math approximation mechanism in the communication complexity model due to Dobzinski, Fu and Kleinberg @cite_15 , and a universally-truthful A mechanism is universally-truthful if, for realization of a the mechanism's coins, each player maximizes his payoff by bidding truthfully. Universally truthful mechanisms are defined formally in Section @math approximation mechanism in the model due to Dobzinski @cite_18 .