Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities
<|reference_start|>Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities: 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.<|reference_end|>
@article{ma2024online,
    title={Online Matching and Contention Resolution for Edge Arrivals with
  Vanishing Probabilities},
    author={Will Ma, Calum MacRury, Pranav Nuti},
    journal={In EC 2024},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.14506},
    primaryClass={cs.DS cs.DM math.CO}
}