Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design

TL;DR

This paper advances budgeted mechanism design by improving approximation ratios for coverage valuations and introducing new polynomial-time mechanisms for XOS problems with independence systems, addressing key challenges in truthful procurement.

Contribution

It provides improved mechanisms for weighted coverage valuations and introduces a general scheme for polynomial-time mechanisms for XOS problems with independence systems.

Findings

Enhanced approximation ratios for weighted coverage valuations.

New polynomial-time mechanisms for XOS problems with independence systems.

Applicable to problems like maximum weighted matchings and 3D-matchings.

Abstract

We study a type of reverse (procurement) auction problems in the presence of budget constraints. The general algorithmic problem is to purchase a set of resources, which come at a cost, so as not to exceed a given budget and at the same time maximize a given valuation function. This framework captures the budgeted version of several well known optimization problems, and when the resources are owned by strategic agents the goal is to design truthful and budget feasible mechanisms, i.e. elicit the true cost of the resources and ensure the payments of the mechanism do not exceed the budget. Budget feasibility introduces more challenges in mechanism design, and we study instantiations of this problem for certain classes of submodular and XOS valuation functions. We first obtain mechanisms with an improved approximation ratio for weighted coverage valuations, a special class of submodular functions that has already attracted attention in previous works. We then provide a general scheme for designing randomized and deterministic polynomial time mechanisms for a class of XOS problems. This class contains problems whose feasible set forms an independence system (a more general structure than matroids), and some representative problems include, among others, finding maximum weighted matchings, maximum weighted matroid members, and maximum weighted 3D-matchings. For most of these problems, only randomized mechanisms with very high approximation ratios were known prior to our results.