  Is perfect matching in NC? That is, is there a deterministic fast parallel
algorithm for it? This has been an outstanding open question in theoretical
computer science for over three decades, ever since the discovery of RNC
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.
  In this paper, we give an NC algorithm for finding a perfect matching in a
planar graph. Our algorithm uses the above-stated fact about counting matchings
in a crucial way. Our main new idea is an NC algorithm for finding a face of
the perfect matching polytope at which $\Omega(n)$ new conditions, involving
constraints of the polytope, are simultaneously satisfied. Several other ideas
are also needed, such as finding a point in the interior of the minimum weight
face of this polytope and finding a balanced tight odd set in NC.
