Parameterized Algorithms for Disjoint Matchings in Weighted Graphs with Applications

It is a well-known and useful problem to find a matching in a given graph G whose size is at most a given parameter k and whose weight is maximized (over all matchings of size at most k in G). In this paper, we consider two natural extensions of this problem. One is to find t disjoint matchings in a given graph G whose total size is at most a given parameter k and whose total weight is maximized, where t is a (small) constant integer. Previously, only the special case where t = 2 was known to be fixed -parameter tractable. In this paper, we show that the problem is fixed parameter tractable for any constant t. When t = 2, the time complexity of the new algorithm is significantly better than that of the previously known algorithm. The other is to find a set of vertex -disjoint paths each of length 1 or 2 in a given graph whose total length is at most a given parameter k and whose total weight is maximized. As interesting applications, we further use the algorithms to speed up several known approximation algorithms (for related NP -hard problems) whose approximation ratio depends on a fixed parameter 0 < epsilon < 1 and whose running time is dominated by the time needed for exactly solving the problems on graphs in which each connected component has at most epsilon(-1) vertices.