Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities

TL;DR

This paper investigates online matching algorithms on random graphs with vanishing edge probabilities, providing new bounds and strategies for different processing orders, including adversarial, random, and prioritized sequences.

Contribution

It introduces a new OCRS with 0.382 selectability under adversarial order, improves upper bounds, and proposes a better scheme with approximately 0.510 selectability when processing order is optimized.

Findings

A 0.382-selectable OCRS for adversarial order.

An upper bound of 0.390 on OCRS selectability.

A 0.5-selectable greedy scheme for random order.

Abstract

We study the performance of sequential contention resolution and matching algorithms on random graphs with vanishing edge probabilities. When the edges of the graph are processed in an adversarially-chosen order, we derive a new OCRS that is $0.382$-selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption. Complementary to this positive result, we show that no OCRS can be more than $0.390$-selectable, significantly improving upon the upper bound of $0.428$ from the literature. We also derive negative results that are specialized to bipartite graphs or subfamilies of OCRS's. Meanwhile, when the edges of the graph are processed in a uniformly random order, we show that the simple greedy contention resolution scheme which accepts all active and feasible edges is $1/2$-selectable. This result is tight due to a known upper bound. Finally, when the algorithm can choose the processing order, we show that a slight tweak to the random order -- give each vertex a random priority and process edges in lexicographic order -- results in a strictly better contention resolution scheme that is $1-\ln(2-1/e)\approx0.510$-selectable. Our positive results also apply to online matching on $1$-uniform random graphs with vanishing (non-identical) edge probabilities, extending and unifying some results from the random graphs literature.