Online Stochastic Matching: Beating 1-1/e

Abstract

We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of 1 − 1 e ≃ 0.632, a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the 1 − 1 e bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this 1− 1 e barrier. Furthermore, we show that no online algorithm can produce a 1−ǫ approximation for an arbitrarily small ǫ for this problem.Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution.At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.

Online Stochastic Matching: New Algorithms with Better Bounds

“…In the stochastic i.i.d. model, when requests are drawn repeatedly and independently from a known probability distribution over the different impression types, Feldman et al [6] prove that one can do better than 1 − 1/e. Under the restriction that the expected number of request of each impression type is an integer, they provide a 0.670-competitive algorithm, later improved by Bahmani and Kapralov [3] to 0.699, and by Manshadi et al…”

When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

“…This question is an extension of similar online stochastic matching questions considered earlier in [10]in that paper, w ij , p ij ∈ {0, 1} and t j = 1. Our model tries to capture the facts that buyers may have a limited attention span (using the timeouts), they might have uncertainties in their preferences (using edge probabilities), and that they might buy the first item they like rather than scanning the entire list.…”

“…In their model, we are given in a first stage probabilistic information about the graph and the cost of the edges is low; in a second stage, the actual graph is revealed but the costs are higher. The original online stochastic matching problem was studied recently by Feldman et al [10]. They gave a 0.67-competitive algorithm, beating the optimal 1 − 1/e-competitiveness known for worst-case models [17,16,21,5,12].…”

“…We get the stochastic online matching problem of Feldman et al [10] if we have w ab = p ab ∈ {0, 1}, in which case we need only consider t b = 1. Their focus was on beating the 1 − 1/e-competitiveness known for worst-case models [17,16,21,5,12]; they gave a 0.67-competitive algorithm that works for the unweighted case with high probability.…”

“…But the algorithm does not knoŵ G (the actual instantiation of the buyers) up front, it only knows G, and hence some more work is required to obtain an algorithm. Further, as was mentioned in the preliminaries, we use OPT to denote the optimal adaptive strategy (instead of the optimal offline matching inĜ as was done in [10]), and compare our algorithm's performance with this OPT.…”