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.

翻译：本文研究了在边概率渐近消失的随机图上，顺序竞争消解与匹配算法的性能。当图中边按对抗性选定顺序处理时，我们提出了一种新的在线竞争消解方案（OCRS），其选择率可达0.382，在边概率渐近消失的假设下达到了文献中的“独立性基准”。与此积极结果相对应，我们证明任何OCRS的选择率均不能超过0.390，显著改进了文献中0.428的上界。我们还针对二分图或OCRS子族给出了专门的否定性结论。另一方面，当图中边按均匀随机顺序处理时，我们证明接受所有活跃可行边的简单贪心竞争消解方案具有1/2选择率，该结果因已知上界而达到紧界。最后，当算法可自主选择处理顺序时，我们通过对随机顺序进行微调——为每个顶点分配随机优先级并按字典序处理边——得到严格更优的竞争消解方案，其选择率为$1-\ln(2-1/e)\approx0.510$。我们的积极结论同样适用于边概率渐近消失（非均匀）的1-均匀随机图上的在线匹配问题，从而扩展并统一了随机图文献中的若干结果。