Two-sided Online Bipartite Matching and Vertex Cover: Beating the Greedy Algorithm

Abstract

We consider the generalizations of two classical problems, online bipartite matching and ski rental, in the field of online algorithms, and establish a novel connection between them. In the original setting of online bipartite matching, vertices from only one side of the bipartite graph are online. Motivated by market clearing applications where both buyers and sellers are online, we study the generalization, called two-sided online bipartite matching, in which all vertices can be online. An algorithm for it should maintain a $$b$$ -matching and try to maximize its size. We show that this problem can be attacked by considering the complementary “dual” problem, two-sided online bipartite vertex cover, which in fact is a generalization of ski rental. As the greedy algorithm is 1/2-competitive for both problems, the challenge is to beat the ratio of 1/2. In this paper, we present new 0.526-competitive algorithms for both problems under the large budget assumption. A key technical ingredient of our results is a charging-based framework for the design and analysis of water-filling type algorithms. This allows us to systematically establish approximation bounds for various water-filling algorithms. On the hardness side, we show that no online randomized algorithm achieves a competitive ratio better than 0.570 and 0.625 respectively for these two problems. Our bounds show that the one-sided optimal ratio of $$1-1/e\approx 0.632$$ is indeed unattainable.