Towards an Optimal Contention Resolution Scheme for Matchings

TL;DR

This paper develops nearly optimal contention resolution schemes for matchings, achieving the best possible balance between probability and selection, and reveals fundamental differences between various scheme types.

Contribution

It introduces asymptotically optimal schemes for general and bipartite matchings, establishing new separations between offline/online and monotone/non-monotone schemes.

Findings

Achieves approximately 0.544 balance for general matchings.

Achieves 0.509 balance for bipartite matchings.

First to demonstrate separation between offline and online schemes, and monotone and non-monotone schemes.

Abstract

In this paper, we study contention resolution schemes for matchings. Given a fractional matching $x$ and a random set $R(x)$ where each edge $e$ appears independently with probability $x_e$, we want to select a matching $M \subseteq R(x)$ such that $\Pr[e \in M \mid e \in R(x)] \geq c$, for $c$ as large as possible. We call such a selection method a $c$-balanced contention resolution scheme. Our main results are (i) an asymptotically (in the limit as $\|x\|_\infty$ goes to 0) optimal $\simeq 0.544$-balanced contention resolution scheme for general matchings, and (ii) a $0.509$-balanced contention resolution scheme for bipartite matchings. To the best of our knowledge, this result establishes for the first time, in any natural relaxation of a combinatorial optimization problem, a separation between (i) offline and random order online contention resolution schemes, and (ii) monotone and non-monotone contention resolution schemes. We also present an application of our scheme to a combinatorial allocation problem, and discuss some open questions related to van der Waerden's conjecture for the permanent of doubly stochastic matrices.