For stationary online bipartite matching, the paper breaks the 1 - 1/e approximation barrier and improves the known competitive ratio to 1 - 1/sqrt(e) + η.
Adaptive policies and approximation schemes for dynamic matching
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Improved Approximations for Stationary Bipartite Matching: Beyond Probabilistic Independence
For stationary online bipartite matching, the paper breaks the 1 - 1/e approximation barrier and improves the known competitive ratio to 1 - 1/sqrt(e) + η.