Matchings in Graphs

We know that counting perfect matchings is polynomial time when we restrict ourselves to the class of planar graphs. Generally speaking, the decision and search versions of a problem turn out to be “easier” than the counting question. For example, the problem of determining if a perfect matching exists, and finding one when it does, is polynomial time in general graphs, while the question of counting all perfect matchings in a graph is #P-complete, and thus may be considered a significantly harder problem. Note that using the decision algorithm as a black box, one might always answer the search question in polynomial time, without looking further into the structure of the graph (iterate over all edges, try to throw them out and query the decision oracle). It is not known, however, whether this inherently sequential strategy can be converted into a parallel algorithm. In the case of planar graphs, we encounter something counter-intuitive — we may count the number of perfect matchings in NC, but it is not known how to find one when it exists in NC. In today’s lecture, we will see how to appeal to the NC counting algorithm as a subroutine to find a perfect matching on the subclass of bipartite graphs (that is, planar bipartite graphs).