In Combinatorial Public Projects, there is a set of projects that may be undertaken, and a set of self-interested players with a stake in the set of projects chosen. A public planner must choose a subset of these projects, subject to a resource constraint, with the goal of maximizing social welfare. Combinatorial Public Projects has emerged as one of the paradigmatic problems in Algorithmic Mechanism Design, a field concerned with solving fundamental resource allocation problems in the presence of both selfish behavior and the computational constraint of polynomial-time. We design a polynomial-time, truthful-in-expectation, (1-1 e)-approximation mechanism for welfare maximization in a fundamental variant of combinatorial public projects. Our results apply to combinatorial public projects when players have valuations that are 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, assuming P != NP. Ours is the first mechanism that achieves a constant factor approximation for a natural NP-hard variant of combinatorial public projects.
Most recently, Dobzinski @cite_11 showed two lower bounds for CPP in the value oracle model: A lower bound of @math on universally truthful mechanisms for flexible CPP with submodular valuations, and a lower bound of @math on truthful-in-expectation mechanisms for CPP with submodular valuations. We note that the latter was the first unconditional lower bound on truthful-in-expectation mechanisms.
@cite_11
2078273056
We show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that provides an approximation ratio of m1 2 -e must use exponentially many value queries, where m is the number of items. In contrast, ignoring incentives there exist constant ratio approximation algorithms for this problem. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far. We demonstrate two additional applications of our new technique: (1) an impossibility result for universally-truthful polynomial time flexible combinatorial public projects and (2) an impossibility result for truthful-in-expectation mechanisms for exact combinatorial public projects. The latter is the first result that bounds the power of polynomial-time truthful in expectation mechanisms in any setting.