Budget Feasible Mechanisms on Matroids

Abstract

Motivated by many practical applications, in this paper we study budget feasible mechanisms where the goal is to procure independent sets from matroids. More specifically, we are given a matroid ${\mathcal {M}}=(E,{\mathcal {I}})$ where each ground (indivisible) element is a selfish agent. The cost of each element (i.e., for selling the item or performing a service) is only known to the element itself. There is a buyer with a budget having additive valuations over the set of elements E. The goal is to design an incentive compatible (truthful) budget feasible mechanism which procures an independent set of the matroid under the given budget that yields the largest value possible to the buyer. Our result is a deterministic, polynomial-time, individually rational, truthful and budget feasible mechanism with 4-approximation to the optimal independent set. Then, we extend our mechanism to the setting of matroid intersections in which the goal is to procure common independent sets from multiple matroids. We show that, given a polynomial time deterministic blackbox that returns $\alpha$-approximation solutions to the matroid intersection problem, there exists a deterministic, polynomial time, individually rational, truthful and budget feasible mechanism with $(3\alpha +1)$-approximation to the optimal common independent set.

This work was partially supported by FET IP project MULTIPEX 317532, polish funds for years 2011–2015 for co-financed international projects, NCN grant UMO-2014/13/B/ST6/00770 and ERC project PAAL-POC 680912. It was also partly supported by the Google Focused Award on Algorithms for Large-scale Data Analysis.