Fully Dynamic Matching in Bipartite Graphs

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.