An $O^*(1.1939^n)$ time algorithm for minimum weighted dominating induced matching

TL;DR

This paper presents an improved exact algorithm with a runtime of O*(1.1939^n) for finding minimum weighted dominating induced matchings in graphs, addressing a known NP-complete problem.

Contribution

It introduces a novel, faster exact algorithm for the minimum weighted dominating induced matching problem with polynomial space complexity.

Findings

Algorithm runs in O*(1.1939^n) time

Improves over previous exact algorithms

Applicable to general graphs

Abstract

Say that an edge of a graph $G$ dominates itself and every other edge adjacent to it. An edge dominating set of a graph $G=(V,E)$ is a subset of edges $E' \subseteq E$ which dominates all edges of $G$. In particular, if every edge of $G$ is dominated by exactly one edge of $E'$ then $E'$ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit it is NP-complete. In this paper we consider the problems of finding a minimum weighted dominating induced matching, if any, and counting the number of dominating induced matchings of a graph with weighted edges. We describe an exact algorithm for general graphs that runs in $O^*(1.1939^n)$ time and polynomial (linear) space. This improves over any existing exact algorithm for the problems in consideration.