Vertex-weighted Online Stochastic Matching with Patience Constraints
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Online Bipartite Matching is a classic problem introduced by Karp, Vazirani, and Vazirani (Proc. ACM STOC, 1990) and motivated by applications such as e-commerce, online advertising, and ride-sharing. We wish to match a set of online vertices (e.g., webpage views, users, customers, or riders) which arrive sequentially one-by-one to a set of offline vertices (e.g., ads, items, or drivers). Each vertex can be matched at most once and the goal is to maximize the matching size. Here, we consider the more general problem allowing vertex weights (on the offline vertices, representing differing item prices, for example), stochastic rewards (attempted matches may be unsuccessful, providing no reward), and patience constraints (each online vertex has a patience, representing the maximum number of unsuccessful match attempts allowed for it). In doing so, we discuss and clarify the definition of competitive ratio in the stochastic setting and compare competing notions. Specifically, we advocate for the use of a competitive ratio that compares the performance of an online algorithm to that of an offline algorithm in the same stochastic setting, whereas some prior work—see, e.g., Mehta and Panigrahi (FOCS, 2012)—uses a linear programming (LP) formulation to compare the performance of an online algorithm to the optimal solution of a closely related non-stochastic offline problem. We define the stochasticity gap for evaluating the usefulness of an LP formulation (which exchanges probabilities for fractional coefficients) of a given stochastic problem and bound this gap for a commonly used LP. We further present and analyze a new algorithm for our problem, achieving a competitive ratio of 0.5, the first constant competitive ratio for this and several related problems.
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