Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities

Will Ma; Calum MacRury; Pranav Nuti

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arXiv · 2406.14506

Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities

Will Ma·Calum MacRury·Pranav Nuti

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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.

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BibTeX RIS

Will Ma, Calum MacRury, Pranav Nuti. 2024-06-20. Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities. https://arxiv.org/abs/2406.14506

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