An Exact 2.9416^n Algorithm for the Three Domatic Number Problem

TL;DR

This paper introduces an exact deterministic algorithm for the three domatic number problem with a runtime of 2.9416^n, improving over the naive 3^n approach, and also offers optimized algorithms for graphs with small maximum degree.

Contribution

The paper presents the first exact algorithm with sub-3^n exponential time for the three domatic number problem, advancing computational methods for this NP-complete problem.

Findings

Deterministic algorithm runs in 2.9416^n time.

Algorithms perform better on graphs with small maximum degree.

Provides both deterministic and randomized algorithms.

Abstract

The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NP-complete, no polynomial-time algorithm is known for it. The naive deterministic algorithm for this problem runs in time 3^n, up to polynomial factors. In this paper, we design an exact deterministic algorithm for this problem running in time 2.9416^n. Thus, our algorithm can handle problem instances of larger size than the naive algorithm in the same amount of time. We also present another deterministic and a randomized algorithm for this problem that both have an even better performance for graphs with small maximum degree.