Complexity results for rainbow matchings

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem. We show, among others, that • max rainbow matching can be approximated by a polynomial algorithm with approximation ratio 2/3 − ε. • max rainbow matching is APX-complete, even when restricted to properly edge-colored linear forests without a 5-vertex path, and is solvable in polynomial time for edge-colored forests without a 4-vertex path. • max rainbow matching is APX-complete, even when restricted to properly edge-colored trees without an 8-vertex path, and is solvable in polynomial time for edge-colored trees without a 7-vertex path. • max rainbow matching is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths. We also address the parameterized complexity of the problem.