Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914967640342528 |
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| author | Ma, Will MacRury, Calum Nuti, Pranav |
| author_facet | Ma, Will MacRury, Calum Nuti, Pranav |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14506 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities Ma, Will MacRury, Calum Nuti, Pranav Data Structures and Algorithms Discrete Mathematics Combinatorics F.2.2; G.2.2 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. |
| title | Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities |
| topic | Data Structures and Algorithms Discrete Mathematics Combinatorics F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2406.14506 |