Second Price Matching with Complete Allocation and Degree Constraints

TL;DR

This paper improves approximation guarantees for the Second Price Matching problem under certain degree constraints and introduces a new variant with expiring goods, providing both hardness results and exact algorithms for specific cases.

Contribution

It presents a new approximation guarantee of 9/10 for (3,2)-regular graphs, an exact polynomial-time algorithm for (d,2)-regular graphs with d≥4, and analyzes a variant with expiring goods.

Findings

Improved approximation guarantee of 9/10 for (3,2)-regular graphs.

Exact polynomial-time algorithm for (d,2)-regular graphs with d≥4.

Hardness of approximation within (1-1/e) for the expiring goods variant.

Abstract

We study the Second Price Matching problem, introduced by Azar, Birnbaum, Karlin, and Nguyen in 2009. In this problem, a bipartite graph (bidders and goods) is given, and the profit of a matching is the number of matches containing a second unmatched bidder. Maximizing profit is known to be APX-hard and the current best approximation guarantee is $1/2$. APX-hardness even holds when all degrees are bounded by a constant. In this paper, we investigate the approximability of the problem under regular degree constraints. Our main result is an improved approximation guarantee of $9/10$ for Second Price Matching in $(3,2)$-regular graphs and an exact polynomial-time algorithm for $(d,2)$-regular graphs if $d\geq 4$. Our algorithm and its analysis are based on structural results in non-bipartite matching, in particular the Tutte-Berge formula coupled with novel combinatorial augmentation methods. We also introduce a variant of Second Price Matching where all goods have to be matched, which models the setting of expiring goods. We prove that this problem is hard to approximate within a factor better than $(1-1/e)$ and show that the problem can be approximated to a tight $(1-1/e)$ factor by maximizing a submodular function subject to a matroid constraint. We then show that our algorithm also solves this problem exactly on regular degree constrained graphs as above.