Towards a Characterization of Truthful Combinatorial Auctions
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We investigate multidimensional covering mechanism-design problems, wherein there are m items that need to be covered and n agents who provide covering objects, with each agent i having a private cost for the covering objects he provides. The goal is to select a set of covering objects of minimum total cost that together cover all the items. We focus on two representative covering problems: uncapacitated facility location (UFL) and vertex cover (VC). For multidimensional UFL, we give a black-box method to transform any Lagrangian-multiplier-preservingρ-approximation algorithm for UFL to a truthful-in-expectation, ρ-approx. mechanism. This yields the first result for multidimensional UFL, namely a truthful-in-expectation 2-approximation mechanism. For multidimensional VC (Multi-VC), we develop a decomposition method that reduces the mechanism-design problem into the simpler task of constructing threshold mechanisms, which are a restricted class of truthful mechanisms, for simpler (in terms of graph structure or problem dimension) instances of Multi-VC. By suitably designing the decomposition and the threshold mechanisms it uses as building blocks, we obtain truthful mechanisms with approximation ratios (n is the number of nodes): (1) O(logn) for Multi-VC on any minor-closed family of graphs; and (2) O(r2logn) for r-dimensional VC on any graph. These are the first truthful mechanisms for Multi-VC with non-trivial approximation guarantees.