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
<ε
<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 ε
-1 vertices.
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