Parameterized Results on Acyclic Matchings with Implications for Related Problems
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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 a positive integer ℓ, Acyclic Matching asks whether G has an acyclic matching of size (i.e., the number of edges) at least ℓ. In this paper, we first 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 the parameter is the size of the matching. Our reduction is general in the sense that it also asserts 𝖥𝖯𝖳-inapproximability for Induced Matching and Uniquely Restricted Matching as well. We also consider three below-guarantee parameters for Acyclic Matching, viz. n/2-ℓ, 𝖬𝖬(𝖦)-ℓ, and 𝖨𝖲(𝖦)-ℓ, where n is the number of vertices in G, 𝖬𝖬(𝖦) is the matching number of G, and 𝖨𝖲(𝖦) is the independence number of G. Furthermore, 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⊆𝖼𝗈𝖭𝖯𝗉𝗈𝗅𝗒.
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