Online Weighted Matching: Beating the $\frac{1}{2}$ Barrier

We study the problem of online weighted bipartite matching in which we want to find a maximum weighted matching between two sets of entities, e.g. matching impressions in online media to advertisers. Karp et al. [14] designed the elegant algorithm Ranking with competitive ratio 1−1/e for the unweighted case. Without the commonly accepted Free Disposal assumption, it is easy to show that no competitive ratio can be achieved in the weighted case. However, under this assumption, it is not hard to show that algorithm Greedy is 1/2 competitive [17, 7], and this is tight for deterministic algorithms. After more than 25 years from the seminal work of Karp et al.[14], it is still an open question whether an online algorithm with competitive ratio better than 1/2 exists or not. We answer this question affirmatively by presenting randomized algorithm StochasticGreedy with competitive ratio greater than 0.501. We also optimize this algorithm and propose slightly changed algorithm OptimizedStochasticGreedy with competitive ratio greater than 0.5018. In light of the hardness result of Kapralov et al. [12] that restricts beating the 1/2-competitive ratio for Monotone Submodular Welfare Maximization problem, our result can be seen as an evidence that solving weighted matching problem is strictly easier than submodular welfare maximization in online settings. Free disposal assumption states that advertisers are either indifferent or happier if they receive more impressions than the capacities stated in their contracts. Advertiser a pays some fixed amount based on its contract to receive Ca (capacity of a) impressions. Assigning more impressions to a only makes it happier since the contract terms do not change and a will not pay more. We can also assume that each advertiser has capacity 1 since we can replace a with Ca identical copies each with capacity 1.