Frugality ratios and improved truthful mechanisms for vertex cover

TL;DR

This paper explores frugality ratios in set-system auctions, introduces a new truthful auction for vertex cover with a 2-approximation, and links local optimality to frugality bounds within a constant factor.

Contribution

It presents a novel truthful polynomial-time auction for vertex cover based on local ratio algorithms and establishes frugality bounds using alternative cost notions.

Findings

The auction guarantees a vertex cover within twice the optimal cost.

Frugality ratios are analyzed with new notions of fair cost.

Local optimality helps derive constant-factor frugality bounds.

Abstract

In {\em set-system auctions}, there are several overlapping teams of agents, and a task that can be completed by any of these teams. The buyer's goal is to hire a team and pay as little as possible. Recently, Karlin, Kempe and Tamir introduced a new definition of {\em frugality ratio} for this setting. Informally, the frugality ratio is the ratio of the total payment of a mechanism to perceived fair cost. In this paper, we study this together with alternative notions of fair cost, and how the resulting frugality ratios relate to each other for various kinds of set systems. We propose a new truthful polynomial-time auction for the vertex cover problem (where the feasible sets correspond to the vertex covers of a given graph), based on the {\em local ratio} algorithm of Bar-Yehuda and Even. The mechanism guarantees to find a winning set whose cost is at most twice the optimal. In this situation, even though it is NP-hard to find a lowest-cost feasible set, we show that {\em local optimality} of a solution can be used to derive frugality bounds that are within a constant factor of best possible. To prove this result, we use our alternative notions of frugality via a bootstrapping technique, which may be of independent interest.