Perfect Matchings in ~O(n^{1.5}) Time in Regular Bipartite Graphs

We consider the well-studied problem of finding a perfect matching in $d$-regular bipartite graphs with $2n$ vertices and $m = nd$ edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes $O(m sqrt{n})$ time, in regular bipartite graphs, a perfect matching is known to be computable in $O(m)$ time. Very recently, the $O(m)$ bound was improved to $O(min{m, frac{n^{2.5}ln n}{d}})$ expected time, an expression that is bounded by $tilde{O}(n^{1.75})$. In this paper, we further improve this result by giving an $O(min{m, frac{n^2ln^3 n}{d}})$ expected time algorithm for finding a perfect matching in regular bipartite graphs; as a function of $n$ alone, the algorithm takes expected time $O((nln n)^{1.5})$. To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite graph with $O(nln n)$ edges that has a perfect matching with high probability. The matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp gives us an $tilde{O}(n^{1.5})$ running time, we present a tighter analysis for our special case that results in the stronger $tilde{O}(min{m, frac{n^2}{d} })$ time mentioned earlier. Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Halls theorem ``witnesses for a perfect matching in a bipartite graph that we prove. We believe this theorem may be of independent interest; as another example application, we show that a perfect matching in the support of an $n times n$ doubly stochastic matrix with $m$ non-zero entries can be found in expected time $tilde{O}(m + n^{1.5})$.