On (Random-order) Online Contention Resolution Schemes for the Matching Polytope of (Bipartite) Graphs

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We present new results for online contention resolution schemes for the matching polytope of graphs, in the random-order (RCRS) and adversarial (OCRS) arrival models. Our results include improved selectability guarantees (i.e., lower bounds), as well as new impossibility results (i.e., upper bounds). By well-known reductions to the prophet (secretary) matching problem, a c-selectable OCRS (RCRS) implies a c-competitive algorithm for adversarial (random order) edge arrivals. Similar reductions are also known for the query-commit matching problem. For the adversarial arrival model, we present a new analysis of the OCRS of Ezra et al.~(EC, 2020). We show that this scheme is 0.344-selectable for general graphs and 0.349-selectable for bipartite graphs, improving on the previous 0.337 selectability result for this algorithm. We also show that the selectability of this scheme cannot be greater than 0.361 for general graphs and 0.382 for bipartite graphs. We further show that no OCRS can achieve a selectability greater than 0.4 for general graphs, and 0.433 for bipartite graphs.
For random-order arrivals, we present two attenuation-based schemes which use new attenuation functions. Our first RCRS is 0.474-selectable for general graphs, and our second is 0.476-selectable for bipartite graphs. These results improve upon the recent 0.45 (and 0.456) selectability results for general graphs (respectively, bipartite graphs) due to Pollner et al.~(EC, 2022). On general graphs, our 0.474-selectable RCRS provides the best known positive result even for offline contention resolution, and also for the correlation gap. We conclude by proving a fundamental upper bound of 0.5 on the selectability of RCRS, using bipartite graphs.