On (Random-order) Online Contention Resolution Schemes for the Matching Polytope of (Bipartite) Graphs
Calum MacRury, Will Ma, Nathaniel Grammel
TL;DR
The paper studies online and random-order contention resolution for matchings in graphs, presenting OCRS and RCRS tailored to adversarial and random edge arrivals across general and bipartite graphs. It develops attenuation-based schemes, leverages the FKG inequality, and leverages a 1-regular input reduction to obtain tight selectability bounds. New results include OCRS bounds of $0.3445$ (general) and $0.349$ (bipartite) for OCRS, and $0.474$ (general) and $0.478$ (bipartite) for RCRS, along with impossibility results such as a $1/2$ upper bound for RCRS in random order. The work substantially improves the state-of-the-art for online accept/reject, probing, and pricing on graphs, and it contributes a unified analytical framework that connects online and offline CRS considerations via reductions and computational verification.
Abstract
Online Contention Resolution Schemes (OCRS's) represent a modern tool for selecting a subset of elements, subject to resource constraints, when the elements are presented to the algorithm sequentially. OCRS's have led to some of the best-known competitive ratio guarantees for online resource allocation problems, with the added benefit of treating different online decisions -- accept/reject, probing, pricing -- in a unified manner. This paper analyzes OCRS's for resource constraints defined by matchings in graphs, a fundamental structure in combinatorial optimization. We consider two dimensions of variants: the elements being presented in adversarial or random order; and the graph being bipartite or general. We improve the state of the art for all combinations of variants, both in terms of algorithmic guarantees and impossibility results. Some of our algorithmic guarantees are best-known even compared to Contention Resolution Schemes that can choose the order of arrival or are offline. All in all, our results for OCRS directly improve the best-known competitive ratios for online accept/reject, probing, and pricing problems on graphs in a unified manner.