On Budget-Feasible Mechanism Design for Symmetric Submodular Objectives

Abstract

We study a class of procurement auctions with a budget constraint, where an auctioneer is interested in buying resources from a set of agents. The auctioneer would like to select a subset of the resources so as to maximize his valuation function, without exceeding his budget. As the resources are owned by strategic agents, our overall goal is to design mechanisms that are truthful, budget-feasible, and obtain a good approximation to the optimal value. Previous results on budget-feasible mechanisms have considered mostly monotone valuation functions. In this work, we mainly focus on symmetric submodular valuations, a prominent class of non-monotone submodular functions that includes cut functions. We begin with a purely algorithmic result, obtaining a \(\frac{2e}{e-1}\)-approximation for maximizing symmetric submodular functions under a budget constraint. We then proceed to propose truthful, budget feasible mechanisms (both deterministic and randomized), paying particular attention on the Budgeted Max Cut problem. Our results significantly improve the known approximation ratios for these objectives, while establishing polynomial running time for cases where only exponential mechanisms were known. At the heart of our approach lies an appropriate combination of local search algorithms with results for monotone submodular valuations, applied to the derived local optima.