Planarizing gadgets for perfect matching do not exist
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
| Hi-index | 0.00 |
In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)−1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K 3,3-minor free graphs (these include all planar graphs as well as many others) in O(n 3.19) worst case time. Our algorithm can also count the number of perfect matchings in K 3,3-minor free graphs in O(n 2.19) time.