Fully Dynamic Matching in Bipartite Graphs

Abstract

We present two fully dynamic algorithms for maximum cardinality matching in bipartite graphs. Our main result is a deterministic algorithm that maintains a $(3/2 + \epsilon )$ approximation in worst-case update time $O(m^{1/4}\epsilon ^{-2.5})$. This algorithm is polynomially faster than all previous deterministic algorithms for any constant approximation, and faster than all previous algorithms (randomized included) that achieve a better-than-2 approximation. We also give stronger results for bipartite graphs whose arboricity is at most $\alpha$, achieving a $(1+ \epsilon )$ approximation in worst-case update time $O(\alpha (\alpha + \log (n)) + \epsilon ^{-4}(\alpha + \log (n)) + \epsilon ^{-6})$, which is $O(\alpha (\alpha + \log n))$ for constant $\epsilon$. Previous results for small arboricity graphs had similar update times but could only maintain a maximal matching (2-approximation). All these previous algorithms, however, were not limited to bipartite graphs.

A. Bernstein—Supported in part by an NSF Graduate Fellowship and a Simons Foundation Graduate Fellowship.

C. Stein—Supported in part by NSF grants CCF-1349602 and CCF-1421161.