# Greedy Approaches to Online Stochastic Matching

Within the context of stochastic probing with commitment, we consider the online stochastic matching problem; that is, the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching (if possible). We consider the competitiveness of online algorithms in the random order input model (ROM), when the offline vertices are weighted. More specifically, we consider a bipartite stochastic graph G = (U,V,E) where U is the set of offline vertices, V is the set of online vertices and G has edge probabilities (p_e)_e ∈ E and vertex weights (w_u)_u ∈ U. Additionally, G has patience values (ℓ_v)_v ∈ V, where ℓ_v indicates the maximum number of edges adjacent to an online vertex v which can be probed. We assume that U and (w_u)_u ∈ U are known in advance, and that the patience, adjacent edges and edge probabilities for each online vertex are only revealed when the online vertex arrives. If any one of the following three conditions is satisfied, then there is a conceptually simple deterministic greedy algorithm whose competitive ratio is 1-1/e. (1) When the offline vertices are unweighted. (2) When the online vertex probabilities are "vertex uniform"; i.e., p_u,v = p_v for all (u,v) ∈ E. (3) When the patience constraint ℓ_v satisfies ℓ_v ∈{[1,|U|} for every online vertex; i.e., every online vertex either has unit or full patience. Setting the probability p_e = 1 for all e ∈ E, the stochastic problem becomes the classical online bipartite matching problem. Our competitive ratios thus generalize corresponding results for the classical ROM bipartite matching setting.

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