Beyond O*(2^n) in domination-type problems

TL;DR

This paper introduces faster algorithms for several NP-complete domination problems, achieving exponential time complexities below the standard 2^n barrier, and provides an approximation scheme for one of them.

Contribution

It presents the first known algorithms for these problems that outperform the naive exponential solution, with specific time bounds and an approximation scheme.

Findings

CAPACITATED DOMINATING SET solved in O(1.89^n)

LARGEST IRREDUNDANT SET solved in O(1.9657^n)

SMALLEST INCLUSION-MAXIMAL IRREDUNDANT SET solved in O(1.999956^n)

Abstract

In this paper we provide algorithms faster than O*(2^n) for several NP-complete domination-type problems. More precisely, we provide: an algorithm for CAPACITATED DOMINATING SET that solves it in O(1.89^n), a branch-and-reduce algorithm solving LARGEST IRREDUNDANT SET in O(1.9657^n) time and a simple iterative-DFS algorithm for SMALLEST INCLUSION-MAXIMAL IRREDUNDANT SET that solves it in O(1.999956^n) time. We also provide an exponential approximation scheme for CAPACITATED DOMINATING SET. All algorithms require polynomial space. Despite the fact that the discussed problems are quite similar to the DOMINATING SET problem, we are not aware of any published algorithms solving these problems faster than the obvious O*(2^n) solution prior to this paper.