Improved Bounds for Online Stochastic Matching

We study the online stochastic matching problem in a form motivated by Internet display advertisement. Recently, Feldman et al. gave an algorithm that achieves 0.6702 competitive ratio, thus breaking through the 1-1/e barrier. One of the questions left open in their work is to obtain a better competitive ratio by generalizing their two suggested matchings (TSM) algorithm to d-suggested matchings (d-SM). We show that the best competitive ratio that can be obtained with the static analysis used in the d-SM algorithm is upper bounded by 0.76, even for the special case of d-regular graphs, thus suggesting that a dynamic analysis may be needed to improve the competitive ratio significantly. We make the first step in this direction by showing that the RANDOM algorithm, which assigns an impression to a randomly chosen eligible advertiser, achieves 1 - e-ddd/d! = 1-O(1/√d) competitive ratio for d-regular graphs, which converges to 1 as d increases. On the hardness side, we improve the upper bound of 0.989 on the competitive ratio of any online algorithm obtained by Feldman et al. to 1-1/(e + e2) ≅ 0.902. Finally, we show how to modify the TSM algorithm to obtain an improved 0.699 approximation for general bipartite graphs.