Parameterized results on acyclic matchings with implications for related problems

Juhi Chaudhary, Meirav Zehavi

Abstract: 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<math><mi is="true">ℓ</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></math>, Acyclic Matching asks whether G has an acyclic matching of size at least ℓ. In this paper, we prove that assuming W[1]⊈FPT<math><mi mathvariant="sans-serif" is="true">W</mi><mo stretchy="false" is="true">[</mo><mn mathvariant="sans-serif" is="true">1</mn><mo stretchy="false" is="true">]</mo><mo is="true">⊈</mo><mi mathvariant="sans-serif" is="true">FPT</mi></math>, there does not exist any FPT<math><mi mathvariant="sans-serif" is="true">FPT</mi></math>-approximation algorithm for Acyclic Matching that approximates it within a constant factor when parameterized by ℓ. Our reduction also asserts FPT<math><mi mathvariant="sans-serif" is="true">FPT</mi></math>-inapproximability for Induced Matching and Uniquely Restricted Matching. We also consider three below-guarantee parameters for Acyclic Matching, viz. n2−ℓ<math><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ℓ</mi></math>, MM(G)−ℓ<math><mrow is="true"><mi mathvariant="sans-serif" is="true">MM</mi><mo stretchy="false" is="true">(</mo><mi mathvariant="sans-serif" is="true">G</mi><mo stretchy="false" is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ℓ</mi></math>, and IS(G)−ℓ<math><mrow is="true"><mi mathvariant="sans-serif" is="true">IS</mi><mo stretchy="false" is="true">(</mo><mi mathvariant="sans-serif" is="true">G</mi><mo stretchy="false" is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ℓ</mi></math>, where n=V(G)<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">V</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></math>, MM(G)<math><mi mathvariant="sans-serif" is="true">MM</mi><mo stretchy="false" is="true">(</mo><mi mathvariant="sans-serif" is="true">G</mi><mo stretchy="false" is="true">)</mo></math> is the matching number, and IS(G)<math><mi mathvariant="sans-serif" is="true">IS</mi><mo stretchy="false" is="true">(</mo><mi mathvariant="sans-serif" is="true">G</mi><mo stretchy="false" is="true">)</mo></math> 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<math><mrow is="true"><mi mathvariant="sans-serif" is="true">NP</mi></mrow><mo is="true">⊆</mo><mrow is="true"><mi mathvariant="sans-serif" is="true">coNP</mi></mrow><mo is="true">/</mo><mrow is="true"><mi mathvariant="sans-serif" is="true">poly</mi></mrow></math>.