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

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.