An Improved Exact Algorithm for the Domatic Number Problem

TL;DR

This paper presents a faster deterministic algorithm for the 3-domatic number problem with a time complexity of 2.695^n, improving previous bounds, and introduces a more efficient randomized algorithm for graphs with bounded degree.

Contribution

It introduces a new deterministic algorithm with improved exponential time complexity and a randomized algorithm for graphs with bounded degree, advancing the computational methods for the domatic number problem.

Findings

Deterministic algorithm runs in time 2.695^n, improving previous 2.8805^n bound.

Randomized algorithm outperforms previous bounds for graphs with degree >= 5.

Combines existing algorithms with novel techniques for better efficiency.

Abstract

The 3-domatic number problem asks whether a given graph can be partitioned intothree dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695^n (up to polynomial factors). This result improves the previous bound of 2.8805^n, which is due to Fomin, Grandoni, Pyatkin, and Stepanov. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Delta(G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe whenever Delta(G) >= 5. Our new randomized algorithm employs Schoening's approach to constraint satisfaction problems.