Truthful Mechanism Design for Multidimensional Covering Problems

We investigate {\em 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 (\vcp). For multidimensional \ufl, we give a black-box method to transform any {\em Lagrangian-multiplier-preserving} $\rho$-approximation algorithm for \ufl to a truthful-in-expectation, $\rho$-approx. mechanism. This yields the first result for multidimensional \ufl, namely a truthful-in-expectation 2-approximation mechanism. For multidimensional \vcp (\mvcp), we develop a {\em decomposition method} that reduces the mechanism-design problem into the simpler task of constructing {\em threshold mechanisms}, which are a restricted class of truthful mechanisms, for simpler (in terms of graph structure or problem dimension) instances of \mvcp. By suitably designing the decomposition and the threshold mechanisms it uses as building blocks, we obtain truthful mechanisms with the following approximation ratios ($n$ is the number of nodes): (1) $O(r^2\log n)$ for $r$-dimensional \vcp; and (2) $O(r\log n)$ for $r$-dimensional \vcp on any proper minor-closed family of graphs (which improves to $O(\log n)$ if no two neighbors of a node belong to the same player). These are the first truthful mechanisms for \mvcp with non-trivial approximation guarantees.