Perfect Matchings in Õ(n) 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 √ 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, n 2.5 lnn d }) expected time, an expression that is bounded by Õ(n). In this paper, we further improve this result by giving an O(min{m, n 2 ln 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((n lnn)). 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(n lnn) edges. The first-stage is a sub-linear time uniform sampling that reduces the size of the input graph while maintaining certain structural properties of the original graph. The second-stage is a non-uniform sampling that takes linear-time (on the reduced graph) and outputs a graph with O(n lnn) edges, while preserving a matching with high probability. This matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp also gives us an Õ(n) running time, we present a tighter analysis for our special case that results in the stronger Õ(min{m, n 2 d }) time mentioned earlier. Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Hall’s 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×n doubly stochastic matrix with m non-zero entries can be found in expected time Õ(m+ n). ∗Departments of Management Science and Engineering and (by courtesy) Computer Science, Stanford University. Email: [email protected]. Research supported by NSF ITR grant 0428868, NSF CAREER award 0339262, and a grant from the Stanford-KAUST alliance for academic excellence. †Institute for Computational and Mathematical Engineering, Stanford University. Email: [email protected]. Research supported by a Stanford Graduate Fellowship. ‡Department of Computer and Information Science, University of Pennsylvania, Philadelphia PA. Email: [email protected]. Supported in part by a Guggenheim Fellowship, an IBM Faculty Award, and by NSF Award CCF-0635084.