Online Bipartite Matching in the Probe-Commit Model
Allan Borodin, Calum MacRury
TL;DR
This work studies online bipartite matching under the probe-commit model with edge uncertainties and per-vertex patience, introducing a configuration LP (LP_new) that relaxes the offline adaptive benchmark. The authors develop efficient algorithms that achieve competitive ratios matching the best known results for related non-probing problems, including 1/2 in adversarial arrivals and 1-1/e in random order arrivals under known i.d. or worst-case models, with improvements across several submodels. A key methodological advance is the LP_new relaxation and its non-adaptive optimum lemma, which enable exact rounding via VertexProbe and reductions to online contention resolution schemes. The results yield a coherent, scalable framework that unifies patience, downward-closed probing constraints, and various arrival orders, while also revealing a tight adaptivity gap of 1-1/e between adaptive and non-adaptive offline benchmarks. Practically, these insights improve understanding and performance guarantees for stochastic matching problems in settings where probing is costly or constrained, with implications for online advertising, kidney exchange, and related applications.
Abstract
We consider the classical online bipartite matching problem in the probe-commit model. In this problem, when an online vertex arrives, its edges must be probed to determine if they exist, based on known edge probabilities. A probing algorithm must respect commitment, meaning that if a probed edge exists, it must be used in the matching. Additionally, each online vertex has a patience constraint which limits the number of probes that can be made to an online vertex's adjacent edges. We introduce a new configuration linear program (LP) which we prove is a relaxation of an optimal offline probing algorithm. Using this LP, we establish the following competitive ratios which depend on the model used to generate the instance graph, and the arrival order of its online vertices: - In the worst-case instance model, an optimal $1/e$ ratio when the vertices arrive in uniformly at random (u.a.r.) order. - In the known independently distributed (i.d.) instance model, an optimal $1/2$ ratio when the vertices arrive in adversarial order, and a $1-1/e$ ratio when the vertices arrive in u.a.r. order. The latter two results improve upon the previous best competitive ratio of $0.46$ due to Brubach et al. (Algorithmica 2020), which only held in the more restricted known i.i.d. (independent and identically distributed) instance model. Our $1-1/e$-competitive algorithm matches the best known result for the prophet secretary matching problem due to Ehsani et al. (SODA 2018). Our algorithm is efficient and implies a $1-1/e$ approximation ratio for the special case when the graph is known. This is the offline stochastic matching problem, and we improve upon the $0.42$ approximation ratio for one-sided patience due to Pollner et al. (EC 2022), while also generalizing the $1-1/e$ approximation ratio for unbounded patience due to Gamlath et al. (SODA 2019).