Perfect Matchings in $O(n\log n)$ Time in Regular Bipartite Graphs

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 bound was improved first to $\tilde O\left(\min\{m, n^{2.5}/d\}\right)$ and then to $\tilde O\left(\min\{m, n^2/d\}\right)$. 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 alternating random walk to successively find augmenting paths. Our algorithm may be viewed as using adaptive uniform sampling, and is thus able to bypass the limitations of (nonadaptive) uniform sampling established in...