Mahyar Salek

Frugal and Truthful Auctions for Vertex Covers, Flows and Cuts

We study truthful mechanisms for hiring a team of agents in three classes of set systems: Vertex Cover auctions, k-???ow auctions, and cut auctions. For Vertex Cover auctions, the vertices are owned by selfish and rational agents, and the auctioneer wants to purchase a vertex cover from them. For k-???ow auctions, the edges are owned by the agents, and the auctioneer wants to purchase k edge-disjoint s-t paths, for given s and t. In the same setting, for cut auctions, the auctioneer wants to purchase an s-t cut. Only the agents know their costs, and the auctioneer needs to select a feasible set and payments based on bids made by the agents. We present constant-competitive truthful mechanisms for all three set systems. That is, the maximum overpayment of the mechanism is within a constant factor of the maximum overpayment of any truthful mechanism, for every set system in the class. The mechanism for Vertex Cover is based on scaling each bid by a multiplier derived from the dominant eigenvector of a certain matrix. The mechanism for k-???ows prunes the graph to be minimally (k + 1)-connected, and then applies the Vertex Cover mechanism. Similarly, the mechanism for cuts contracts the graph until all s-t paths have length exactly 2, and then applies the Vertex Cover mechanism.

Envy-Free Allocations for Budgeted Bidders

We study the problem of identifying prices to support a given allocation of items to bidders in an envy-free way. A bidder will envy another bidder if she would prefer to obtain the other bidders item at the price paid by that bidder. Envy-free prices for allocations have been studied extensively; here, we focus on the impact of budgets: beyond their willingness to pay for items, bidders are also constrained by their ability to pay, which may be lower than their willingness. In a recent paper, Aggarwal et al. show that a variant of the Ascending Auction finds a feasible and bidder-optimal assignment and supporting envy-free prices in polynomial time so long as the input satisfies certain non-degeneracy conditions. While this settles the problem of finding a feasible allocation, an auctioneer might sometimes also be interested in a specific allocation of items to bidders. We present two polynomial-time algorithms for this problem, one which finds maximal prices supporting the given allocation (if such prices exist), and another which finds minimal prices. We also prove a structural result characterizing when different allocations are supported by the same minimal price vector.