Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs

• Authors: Mohammad Mahdian;Qiqi Yan
• Venue: Proceedings of the forty-third annual ACM symposium on Theory of computing
• Year: 2011

In a seminal paper, Karp, Vazirani, and Vazirani show that a simple ranking algorithm achieves a competitive ratio of 1-1/e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1-1/e is also shown to be optimal. Their result also implies that in the random arrivals model defined by Goel and Mehta, where the online nodes arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1-1/e. In this paper, we study the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1-1/e ≈ 0.632 barrier in the adversarial model. Our result also extends to the i.i.d. distribution model of Feldman et al., removing the assumption that the distribution is known. Our analysis has two main steps. First, we exploit certain dominance and monotonicity properties of the ranking algorithm to derive a family of factor-revealing linear programs (LPs). In particular, by symmetry of the ranking algorithm in the random arrivals model, we have the monotonicity property on both sides of the bipartite graph, giving good "strength" to the LPs. Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we introduce the technique of strongly factor-revealing LPs. In particular, we derive a family of modified LPs with similar strength such that the optimal value of any single one of these new LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis.