Exact Algorithms for Dominating Induced Matching Based on Graph Partition

TL;DR

This paper presents a new exact algorithm with improved exponential time complexity for finding a dominating induced matching in a graph, a problem known to be NP-hard, by leveraging structural properties and graph partitioning techniques.

Contribution

It introduces a faster exact algorithm with time complexity $1.1467^n n^{O(1)}$ for the NP-hard dominating induced matching problem, improving previous results.

Findings

Achieved a $1.1467^n n^{O(1)}$-time algorithm for the problem

Identified structural properties and 'good vertices' to optimize the search process

Provided theoretical analysis and proof of algorithm efficiency

Abstract

A dominating induced matching, also called an efficient edge domination, of a graph $G=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges is a subset $F \subseteq E$ of edges in the graph such that no two edges in $F$ share a common endpoint and each edge in $E\setminus F$ is incident with exactly one edge in $F$. It is NP-hard to decide whether a graph admits a dominating induced matching or not. In this paper, we design a $1.1467^nn^{O(1)}$-time exact algorithm for this problem, improving all previous results. This problem can be redefined as a partition problem that is to partition the vertex set of a graph into two parts $I$ and $F$, where $I$ induces an independent set (a 0-regular graph) and $F$ induces a perfect matching (a 1-regular graph). After giving several structural properties of the problem, we show that the problem always contains some "good vertices", branching on which by including them to either $I$ or $F$ we can effectively reduce the graph. This leads to a fast exact algorithm to this problem.