Simple Random Order Contention Resolution for Graphic Matroids with Almost no Prior Information

TL;DR

This paper introduces a simple, constant-selectable random order contention resolution scheme for graphic matroids that requires minimal prior information, specifically only the size of the ground set, and works even under adversarial order with sampling.

Contribution

It presents the first strong ROCRS for graphic matroids that needs almost no prior information, only the ground set size, and functions under adversarial order with sampling.

Findings

Provides a simple constant-selectable ROCRS for graphic matroids.

Works under adversarial order with sampling.

Requires only the size of the ground set as prior knowledge.

Abstract

Random order online contention resolution schemes (ROCRS) are structured online rounding algorithms with numerous applications and links to other well-known online selection problems, like the matroid secretary conjecture. We are interested in ROCRS subject to a matroid constraint, which is among the most studied constraint families. Previous ROCRS required to know upfront the full fractional point to be rounded as well as the matroid. It is unclear to what extent this is necessary. Fu, Lu, Tang, Turkieltaub, Wu, Wu, and Zhang (SOSA 2022) shed some light on this question by proving that no strong (constant-selectable) online or even offline contention resolution scheme exists if the fractional point is unknown, not even for graphic matroids. In contrast, we show, in a setting with slightly more knowledge and where the fractional point reveals one by one, that there is hope to obtain strong ROCRS by providing a simple constant-selectable ROCRS for graphic matroids that only requires to know the size of the ground set in advance. Moreover, our procedure holds in the more general adversarial order with a sample setting, where, after sampling a random constant fraction of the elements, all remaining (non-sampled) elements may come in adversarial order.