Budget Feasible Mechanisms on Matroids

Stefano Leonardi,Piotr Sankowski,Qiang Zhang,Gianpiero Monaco

doi:10.1007/s00453-020-00781-9

Stefano Leonardi, Piotr Sankowski + Show 2 more

Open Access

https://doi.org/10.1007/s00453-020-00781-9

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Abstract

Motivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid {mathcal {M}}=(E,{mathcal {I}}). Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair {mathcal {M}}=(E,{mathcal {I}}) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an alpha -approximation to the problem of finding a maximum-value independent set in {mathcal {M}}, there exists an individually rational, truthful and budget feasible mechanism which is (3alpha +1)-approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids.

Highlights

Procurement auctions (a.k.a. reverse auctions) are executed by governments or private companies to purchase commodities and services from providers
The goal of our work is to design budget feasible mechanisms for procuring items that form an independent set in a given matroid structure
To the best of our knowledge, this is the first time that matroid constraints are considered in budget feasible mechanism design

Summary

Introduction

Procurement auctions (a.k.a. reverse auctions) are executed by governments or private companies to purchase commodities and services from providers. The budget that can be spent in a procurement auction is often limited imposing a limit on the total payments that can be handled to the providers. Motivated by the setting described above, in this work we study the problem of a budget limited buyer with additive valuations on a set of indivisible items, each item controlled from an independent strategic agent. We consider the case of a buyer that is constrained to purchase a subset of objects that forms an independent set of an underlying matroid structure. Matroids are linked to many interesting economic applications, e.g., auctions [5, 12, 18], spectrum market [24], scheduling [11] and house market [19]

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