Extending the Kernel for Planar Steiner Tree to the Number of Steiner Vertices

Ondřej Suchý

doi:10.1007/s00453-016-0249-1

Abstract

In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk et al. (55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, 2014) gave a polynomial kernel for Steiner Tree in planar graphs and graphs of bounded genus, when parameterized by $$|T|+k$$ , the total number of vertices in the constructed subgraph. In this paper we present several polynomial time applicable reduction rules for Steiner Tree in graphs of bounded genus. In an instance reduced with respect to the presented reduction rules, the number of terminals |T| is at most cubic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in graphs of bounded genus for the parameterization by the number k of Steiner vertices in the solution. We give better bounds for Steiner Tree in planar graphs.

Highlights

The Steiner problem is a classical problem of theoretical computer science and a fundamental problem of network design
By folklore result Planar Steiner Tree is known to be fixed parameter tractable with respect to this parameter, it was not known whether there is a polynomial kernel. We resolve this question as follows: We present several polynomial time reduction rules and show that if the rules are exhaustively applied the number of terminals is at most quadratic in k
First Pilipczuk et al [35] showed that there is a subexponential algorithm for Planar Steiner Tree with respect to this parameterization running in O(2√O(((k+|T |) log(k+|T |))2/3)n) time

Summary

Introduction

The Steiner problem is a classical problem of theoretical computer science and a fundamental problem of network design. The problem was recently studied with respect to the parameter “the number of edges in an optimal Steiner tree” or equivalently |T | + k by Pilipczuk, Pilipczuk, Sankowski, and van Leeuwen [35, 36], who obtained a subexponential algorithm [35] and a polynomial kernel [36] for the problem with respect to this parameterization In particular they proved the following proposition. The number of edges of an optimal Steiner tree of that instance is at most quadratic in k and we can use the algorithm of Proposition 1 to obtain a polynomial kernel This improves the result of Pilipczuk et al qualitatively, since we give a polynomial kernel with respect to a parameter that is always smaller and can be arbitrarily smaller than the parameter they use. It improves it quantitatively, as our rules never increase the number of edges in an optimal Steiner tree, and, the kernel obtained by first running our rules is always at most as big as the one obtained by starting directly with the algorithm of Proposition 1

Related Work

Algorithm

General Reduction Rules

Plane Specific Reduction Rules

Main Theorem and Time Complexity

Conclusion and Future Directions