New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs

TL;DR

This paper introduces new geometric representations for tolerance and multitolerance graphs, enabling the authors to resolve the complexity of the dominating set problem, proving it polynomial-time solvable for tolerance graphs and APX-hard for multitolerance graphs.

Contribution

The paper presents two novel geometric representations for tolerance and multitolerance graphs, leading to new complexity results for the dominating set problem.

Findings

Polynomial-time algorithm for dominating set on tolerance graphs.

APX-hardness of dominating set on multitolerance graphs.

First known complexity difference between these two graph classes.

Abstract

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed for each interval - one from the left and one from the right side. Several efficient algorithms for optimization problems that are NP-hard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complexity status of some fundamental algorithmic problems on tolerance and multitolerance graphs, such as the dominating set problem, remained unresolved until now, three decades after the introduction of tolerance graphs. In this article we introduce two new geometric representations for tolerance and multitolerance graphs, given by points and line segments in the plane. Apart from being important on their own, these new representations prove to be a powerful tool for deriving both hardness results and polynomial time algorithms. Using them, we surprisingly prove that the dominating set problem can be solved in polynomial time on tolerance graphs and that it is APX-hard on multitolerance graphs, solving thus a longstanding open problem. This problem is the first one that has been discovered with a different complexity status in these two graph classes.