Moderately exponential time algorithms for the maximum induced matching problem

Abstract

An induced matching $M\subseteq E$ in a graph $G=(V, E)$ is a matching such that no two edges in $M$ are joined by any third edge of the graph. The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. It is NP-hard. Branch-and-reduce algorithms proposed in the previous results for the Maximum Induced Matching problem use a standard branching rule: for a vertex $v$, it branches into $deg(v)+1$ subproblems that either $v$ is not an endvertex of any edge in $M$ or $v$ and one of its neighbor are endvertices of an edge in $M$. In this paper, we give a simple branch-and-reduce algorithm consisting of a boundary condition, a reduction rule, and a branching rule. Especially, the branching rule only branches the original problem into two subproblems. When the algorithm meets the input instance satisfying the boundary condition, we reduce the Maximum Induced Matching problem to the Maximum Independent Set problem. By using the measure-and-conquer approach to analyze the running time of the algorithm, we show that this algorithm runs in time $O^{*}(1.4658^n)$ which is faster than previously known algorithms. By adding two branching rules in the simple exact algorithm, we obtain an $O^{*}(1.4321^n)$-time algorithm for the Maximum Induced Matching problem. Moreover, we give a moderately exponential time $\rho$-approximation algorithm, $0 < \rho < 1$, for the Maximum Induced Matching problem. For $\rho =0.5$, the algorithm runs in time $O^{*}(1.3348^n)$.