PREPRINT

# Towards an Optimal Contention Resolution Scheme for Matchings

Pranav Nuti and Jan Vondrák

Submitted on 7 November 2022

## Abstract

In this paper, we study contention resolution schemes for matchings. Given a fractional matching $x$ and a random set $R\left(x\right)$ where each edge $e$ appears independently with probability ${x}_{e}$, we want to select a matching $M\subseteq R\left(x\right)$ such that $Pr\left[e\in M\mid e\in R\left(x\right)\right]\ge c$, 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 $‖x{‖}_{\mathrm{\infty }}$ 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.

## Preprint

Comment: 22 pages

Subjects: Computer Science - Data Structures and Algorithms; Computer Science - Discrete Mathematics; Mathematics - Probability