Seeking a Vertex of the Planar Matching Polytope in NC

Abstract

AbstractFor 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 \(\mathbb{{\cal P}}(G)\) 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.KeywordsPlanar GraphPerfect MatchSimple CycleClosed WalkPlanar MatchThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.