The Steiner cycle and path cover problem on interval graphs

Ante Ćustić,Stefan Lendl

doi:10.1007/s10878-021-00757-7

Abstract

The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

Highlights

In this paper we investigate the Steiner cycle and Steiner path problem on interval graphs
In this work we generalize the algorithms of Manacher et al (1990) to the Steiner setting and obtain first linear time algorithms for the Steiner path cover and Steiner cycle problem on interval graphs
We show that the basic greedy principle, that is at the core of efficient algorithms for the path cover problem on interval graphs, can be generalized by the introduction of neglectable intervals

Summary

Introduction

In this paper we investigate the Steiner cycle and Steiner path problem on interval graphs. To our knowledge, these problems have not been studied for this specific graph class. Journal of Combinatorial Optimization are special cases of the Steiner variants, are extensively studied for interval graphs and can be solved in linear time, if the intervals are given as an endpoint sorted list (Hung and Chang 2011; Keil 1985; Arikati and Rangan 1990; Manacher et al 1990). In this work we generalize the algorithms of Manacher et al (1990) to the Steiner setting and obtain first linear time algorithms for the Steiner path cover and Steiner cycle problem on interval graphs. To obtain our results we extend the tools introduced in Hung and Chang (2011), and prove an elegant skipping lemma for the Steiner setting

Definitions and preliminary results

The Steiner path cover problem

The Steiner cycle problem

Conclusion