Almost Exact Matchings

• Authors: Raphael Yuster
• Venue: APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
• Year: 2007

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 Ghas a perfect matching, exactly kedges of which are red. More generally if the matching number of Gis m= m(G), the goal is to find a matching with medges, exactly kedges 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 kred edges, or exhibits a matching with m(G) 茂戮驴 1 edges having exactly kred 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 K3,3-minor free graphs (these include all planar graphs as well as many others) in O(n3.19) worst case time. Our algorithm can also count the number of perfect matchings in K3,3-minor free graphs in O(n2.19) time.