A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set

TL;DR

This paper presents a new exact algorithm for finding a minimum independent dominating set in a graph, significantly improving the previous best exponential time algorithms and establishing a near-tight lower bound.

Contribution

It introduces the first non-trivial algorithm with a running time of O(1.3569^n) for the problem, improving upon prior methods that enumerated all maximal independent sets.

Findings

New algorithm with O(1.3569^n) running time

Lower bound of Ω(1.3247^n) on worst-case complexity

Almost tight analysis of the algorithm's efficiency

Abstract

An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of \Omega(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.