Motivated by applications from ride-sharing and kidney exchange, we study the problem of matching agents who arrive at a marketplace over time and leave after d time periods. Agents can only be matched while they are present in the marketplace. Each pair of agents can yield a different match value, and the planner's goal is to maximize the total value over a finite time horizon. First we study the case in which vertices arrive in an adversarial order. We provide a randomized 1 /4-competitive

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... orithm building on a result by Feldman et al. [2009b] and Lehmann et al. [2006] . We extend the model to the case in which departure times are drawn independently from a distribution with non-decreasing hazard rate, for which we establish a 1 /8-competitive algorithm. When the arrival order is chosen uniformly at random, we show that a batching algorithm, which computes a maximum-weighted matching every (d + 1) periods, is 0.279-competitive.