## 7 October 2022

# Perfect Matchings in O(n log n) time in Regular Bipartite Graphs

## Nikolas Patris

Abstract: In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m = nd edges. The best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time \(O(m \sqrt{n})\). In regular bipartite graphs, however, a matching is known to be computable in \(O(m)\) time (due to Cole, Ost and Schirra). In a recent line of work by Goel, Kapralov and Khanna the \(O(m)\) time algorithm was improved first to O˜ min{m, n2.5/d} and then to O˜ min{m, n2/d} . It was also shown that the latter algorithm is optimal up to polylogarithmic factors among all algorithms that use non-adaptive uniform sampling to reduce the size of the graph as a first step. In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in \(O(n \log n)\) time (both in expectation and with high probability). The algorithm performs an appropriately truncated random walk on a modified graph to successively find augmenting paths. Our algorithm may be viewed as using adaptive uniform sampling, and is thus able to bypass the limitations of (non-adaptive) uniform sampling established in earlier work. We also show that randomization is crucial for obtaining \(o(nd)\) time algorithms by establishing an \(\Omega(nd)\) lower bound for any deterministic algorithm. Our techniques also give an algorithm that successively finds a matching in the support of a doubly stochastic matrix in expected time \(O(n \log_2 n)\) time, with O(m) pre-processing time; this gives a simple \(O(m + mn \log_2 n)\) time algorithm for finding the Birkhoff-von Neumann decomposition of a doubly stochastic matrix.
Authors: Ashish Goel, Michael Kapralov, Sanjeev Khanna