Frugal Auction Design for Set Systems: Vertex Cover and Knapsack

We study mechanism design for procurement auctions in which the goal is to buy a subset of items or hire a team of providers. In order to measure the efficiency of a mechanism, one defines an appropriate benchmark which denotes a reasonable expectation of the payments and defines the overpayment of a mechanism based on the benchmark. This ratio is called the \em frugality ratio of the mechanism. Procurement auctions are well-studied and benchmarks proposed for these auctions have evolved over a sequence of papers ~\citearcher2007frugal,chen2010frugal,elkind2007frugality,karlinbeyond,kempe2010frugal. In this work, we introduce a newer benchmark, and based on that, study classic procurement auctions. Our benchmark addresses critical issues raised by the unintuitive behavior of the previous benchmarks. We show two attractive properties for our benchmark which have been lacking in the previous proposals: \em monotonicity and \em smoothness. Based on our benchmark, we provide positive results for vertex cover and knapsack auctions. Prior to this work, \kempefrugal\citekempe2010frugal propose a constant approximation mechanism for vertex cover auctions. However, their analysis suffers from an error. We give a correct analysis to the mechanism of \kempefrugal\citekempe2010frugal with respect to our benchmark. In particular, we prove their mechanism is optimal up to a constant factor. Our analysis is different from what \kempefrugal\citekempe2010frugal propose. We also study the knapsack auctions and give a truthful mechanism for such auctions with a bounded frugality ratio. We show that this is almost tight by presenting a lower bound on the frugality ratio of any truthful mechanism for such auctions. All our results depend on both properties of the benchmark.