Edge-Weighted Online Bipartite Matching

TL;DR

This paper introduces the first online algorithm for edge-weighted bipartite matching that surpasses the 1/2 competitive ratio barrier, using a novel correlated selection technique, indicating the problem's relative simplicity compared to submodular welfare maximization.

Contribution

The paper presents a new online algorithm with a competitive ratio of at least 0.5086 for edge-weighted bipartite matching, breaking the long-standing 1/2 barrier, and introduces the online correlated selection subroutine.

Findings

Achieves a competitive ratio of at least 0.5086 for edge-weighted bipartite matching.

Introduces the online correlated selection (OCS) technique for online decision-making.

Suggests edge-weighted bipartite matching is easier than submodular welfare maximization online.

Abstract

Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal competitive ratio of $1-1/e$. Later, Aggarwal et al. (SODA 2011) generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial $1/2$-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing $1/2$ barrier and achieves a competitive ratio of at least $0.5086$. In light of the hardness result of Kapralov, Post, and Vondr\'ak (SODA 2013) that restricts beating a $1/2$ competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.