Black-box reductions for cost-sharing mechanism design

Abstract

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 minimum-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(α log n)-approximation where the prices recover the cost incurred. Thus, we decouple the task of truthfully computing an outcome with near-optimal social cost from the cost-sharing problem. This is fruitful since truthful mechanism-design, especially for single-dimensional problems, is a relatively well-understood and manageable task. Our second reduction nicely complements the first one by showing that any LP-based 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 find a slew of applications, yielding, as corollaries, the first or improved polytime cost-sharing mechanisms for a variety of problems. For example, our first reduction coupled with the celebrated VCG mechanism shows that for any cost-sharing problem (with a monotone cost function) one can obtain a truthful mechanism that achieves an O(log n)-approximation where the prices recover the cost incurred. Other applications include O(log n)-approximation mechanisms for: survivable network design problems, facility location (FL) problems including capacitated and connected FL problems, and minimum-makespan scheduling on unrelated machines. Our results demonstrate that in contrast with our current understanding of group-strategyproof and acyclic mechanisms, strategyproofness allows for ample flexibility in cost-sharing mechanism design enabling one to effectively leverage various algorithmic results.