Vijay V. Vazirani, University of California, Irvine
3 January, 2018
Is matching in NC, i.e., is there a deterministic fast parallel
algorithm for it? This has been an outstanding open question in TCS for
over three decades, ever since the discovery of Random NC matching
algorithms. Within this question, the case of planar graphs has remained an
enigma: On the one hand, counting the number of perfect matchings is far
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 enough, 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 Nima Anari.