Random sampling in cut, flow, and network design problems
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Randomized algorithms
Bipartite Edge Coloring in $O(\Delta m)$ Time
SIAM Journal on Computing
Using randomized sparsification to approximate minimum cuts
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Switch Scheduling via Randomized Edge Coloring
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Maximum Matchings via Gaussian Elimination
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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In this article we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first nontrivial algorithm, with running time O(mn), dates back to König's work in 1916 (here m=nd is the number of edges in the graph, 2n is the number of vertices, and d is the degree of each node). The currently most efficient algorithm takes time O(m), and is due to Cole et al. [2001]. We improve this running time to O(min{m, n2.5ln n/d}); this minimum can never be larger than O(n1.75&sqrt;ln n). We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a d-regular bipartite graph independently with a probability p = O(n ln n/d2) then the resulting graph has a perfect matching with high probability. The proof involves a decomposition of the graph into pieces which are guaranteed to have many perfect matchings but do not have any small cuts. We then establish a correspondence between potential witnesses to nonexistence of a matching (after sampling) in any piece and cuts of comparable size in that same piece. Karger's sampling theorem [1994a, 1994b] for preserving cuts in a graph can now be adapted to prove our uniform sampling theorem for preserving perfect matchings. Using the O(m&sqrt;n) algorithm (due to Hopcroft and Karp [1973]) for finding maximum matchings in bipartite graphs on the sampled graph then yields the stated running time. We also provide an infinite family of instances to show that our uniform sampling result is tight up to polylogarithmic factors (in fact, up to ln2 n).