Abtract: Is matching in NC? In other words, is there a deterministic fast parallel algorithm for it? This has been an open question for over three decades, ever since the discovery of Random NC matching algorithms. Within this question, the case of planar graphs had remained an enigma: On one hand, counting the number of perfect matchings is generally believed to be harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution.
The case of bipartite planar graphs was solved by Miller and Naor in 1989 via a flow-based algorithm. In 2000, Mahajan and Varadarajan gave an elegant way of using counting matchings to finding one, hence giving a different NC algorithm.
However, non-bipartite planar graphs still didn't yield: the stumbling block being odd tight cuts. Interestingly, these are also a key to the solution: a balanced odd tight cut leads to a straight-forward divide and conquer NC algorithm. The remaining task is to find such a cut in NC. This requires several algorithmic ideas, such as finding a point in the interior of the minimum weight face of the perfect matching polytope and uncrossing odd tight cuts.
Paper available at: https://arxiv.org/pdf/1709.07822.pdf
Joint work with Vijay Vazirani.