Online Edge-Weighted Matching Beyond 0.55

Abstract

We give a $0.556195$-competitive randomized algorithm for edge-weighted online bipartite matching with free disposal, improving the $0.5368$ guarantee of Blanc and Charikar. The result holds for arbitrary nonnegative edge weights and any input sequence fixed independently of the algorithm's randomness. Our algorithm builds on online correlated selection, introduced by Fahrbach, Huang, Tao, and Zadimoghaddam, and its multiway extension due to Blanc and Charikar. Our main contribution is a state-dependent primal--dual analysis of correlated selection. At each weight threshold, we track the state of the selection process jointly with an auxiliary failure event. The analysis accounts for arrivals below the threshold, which change the state without improving the matching value at that threshold. A potential function bounds the dual increase in both cases. We instantiate the analysis with an explicit win map that assigns a common score to an initial range of states. Its selection inequalities are verified by a finite computer-assisted proof covering every required candidate multiplicity and every real threshold; the verifiers are available in an accompanying repository. Triangular unit-weight inputs asymptotically attain the guarantee of the underlying ideal algorithm, proving that its analysis is tight.