Seeking a Vertex of the Planar Matching Polytope in NC

For planar graphs, counting the number of perfect matchings (and hence determining whether there exists a perfect matching) can be done in NC [4, 10]. For planar bipartite graphs, finding a perfect matching when one exists can also be done in NC [8, 7]. However in general planar graphs (when the bipartite condition is removed), no NC algorithm for constructing a perfect matching is known. We address a relaxation of this problem. We consider the fractional matching polytope P(G) of a planar graph G. Each vertex of this polytope is either a perfect matching, or a half-integral solution: an assignment of weights from the set {0, 1/2, 1} to each edge of G so that the weights of edges incident on each vertex of G add up to 1 [6]. We show that a vertex of this polytope can be found in NC, provided G has at least one perfect matching to begin with. If, furthermore, the graph is bipartite, then all vertices are integral, and thus our procedure actually finds a perfect matching without explicitly exploiting the bipartiteness of G.