Parameterized results on acyclic matchings with implications for related problems

A matching M in a graph G is an acyclic matching if the subgraph of G induced by the endpoints of the edges of M is a forest. Given a graph G and ℓ∈N, ACYCLIC MATCHING asks whether G has an acyclic matching of size at least ℓ. In this paper, we prove that assuming W[1]⊈FPT, there does not exist any FPT-approximation algorithm for ACYCLIC MATCHING that approximates it within a constant factor when parameterized by ℓ. Our reduction also asserts FPT-inapproximability for INDUCED MATCHING and UNIQUELY RESTRICTED MATCHING. We also consider three below-guarantee parameters for ACYCLIC MATCHING, viz. [Formula presented], MM(G)−ℓ, and IS(G)−ℓ, where n=V(G), MM(G) is the matching number, and IS(G) is the independence number of G. Also, we show that ACYCLIC MATCHING does not exhibit a polynomial kernel with respect to vertex cover number (or vertex deletion distance to clique) plus the size of the matching unless NP⊆coNP/poly.