On the Induced Matching Problem in Hamiltonian Bipartite Graphs

TL;DR

This paper investigates the computational difficulty of finding maximum induced matchings in Hamiltonian bipartite graphs, establishing hardness results and inapproximability bounds under common complexity assumptions.

Contribution

It proves W[1]-hardness of the problem given a Hamiltonian cycle and establishes strong inapproximability results for maximum induced matchings in Hamiltonian bipartite graphs.

Findings

W[1]-hardness of induced matching problem in Hamiltonian bipartite graphs

Inapproximability within ratio n^{1-psilon} unless NP=P

NP-hardness of approximation within ratio n^{1/4-psilon} in bipartite case

Abstract

In this paper, we study the parameterized complexity and inapproximability of the {\sc Induced Matching} problem in hamiltonian bipartite graphs. We show that, given a hamiltonian cycle in a hamiltonian bipartite graph, the problem is W[1]-hard and cannot be solved in time $n^{o(k^{\frac{1}{2}})}$ unless W[1]=FPT, where $n$ is the number of vertices in the graph. In addition, we show that unless NP=P, the maximum induced matching in a hamiltonian graph cannot be approximated within a ratio of $n^{1-\epsilon}$, where $n$ is the number of vertices in the graph. For a bipartite hamiltonian graph in $n$ vertices, it is NP-hard to approximate its maximum induced matching based on a hamiltonian cycle of the graph within a ratio of $n^{\frac{1}{4}-\epsilon}$, where $n$ is the number of vertices in the graph and $\epsilon$ is any positive constant.