For bipartite graphs (consider only friendships between men and women), we have had fast matching algorithms since the 1950's via augmenting paths. In the 1965 classic paper, Path, Trees and Flowers, Jack Edmonds gives a polynomial-time algorithm for matching on general graphs. This paper also laid out an argument for polynomial-time as efficient computation that would lead to the complexity class P (of P v NP fame).

After Razborov showed that the clique problem didn't have polynomial-size monotone circuits, his proof techniques also showed that matching didn't have polynomial-size monotone circuits and Raz and Wigderson show that matching requires exponential size and linear depth. Because of Edmond's algorithm matching does have polynomial-size circuits in general. NOTs are very powerful.

Can one solve matching in parallel, say the class NC (Nick's class after Pippenger) of problems computable by a polynomial number of processors in polylogarithmic time? Karp, Upfal and Wigderson give a randomized NC algorithm for matching. Mulmuley, Vazirani and Vazirani prove an isolation lemma that allows a randomized reduction of matching to the determinant. Howard Karloff exhibited a Las Vegas parallel algorithm, i.e., never makes a mistake and runs in expected polylogarithmic time.

Can one remove the randomness? An NC algorithm for matching remains elusive but this year brought two nice results in that direction. Ola Svensson and Jakub Tarnawski give a quasi-NC algorithm for general graph matching. Quasi-NC means a quasipolynomial (2

^{polylog}) number of processors. Nima Anari and Vijay Vazirani give an NC algorithm for matching on planar graphs.

Matching is up there with primality, factoring, connectivity, graph isomorphism, satsfiability and the permanent as fixed algorithm problems that have played such a large role in helping us understand complexity. Thanks matching problem and may you find NC nirvana in the near future.

You forgot to mention the LP story where the bipartite case is easy but the extension complexity of general matching is exponential

ReplyDeleteThe Svensson and Tarnawski result was preceded by a Quasi-NC algorithm for bipartite matching by Fenner, Gurjar, & Thierauf in STOC 2916 https://arxiv.org/abs/1601.06319

ReplyDeleteYes, the Fenner, Gurjar & Thierauf paper is a real gem. One of few results with such a beautiful and clean proof so that there is a chance to understand it during a 20-min STOC talk. It was a big inspiration for our work.

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