Black-box reductions for cost-sharing mechanism design

We consider the design of strategyproof cost-sharing mechanisms. We give two simple, but extremely versatile, black-box reductions, that in combination reduce the cost-sharing mechanism-design problem to the algorithmic problem of finding a min-cost solution for a set of players. Our first reduction shows that any truthful, α-approximation mechanism for the social-cost minimization (SCM) problem satisfying a technical no-bossiness condition can be morphed into a truthful mechanism that achieves an O(αlog⁡n)-approximation where the prices recover the cost incurred. Thus, we decouple the approximation and cost-recovery objectives. Our second reduction nicely complements the first one by showing that any LP-relative ρ-approximation for the problem of finding a min-cost solution for a set of players yields a truthful, no-bossy, (ρ+1)-approximation for the SCM problem (and hence, a truthful (ρ+1)log⁡n-approximation cost-sharing mechanism). These reductions yield the first, or improved, polytime cost-sharing mechanisms for a variety of problems.