Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem
Abstract
We study Steiner Forest on -subgraph-free graphs, that is, graphs that do not contain some fixed graph as a (not necessarily induced) subgraph. We are motivated by a recent framework that completely characterizes the complexity of many problems on -subgraph-free graphs. However, in contrast to e.g. the related Steiner Tree problem, Steiner Forest falls outside this framework. Hence, the complexity of Steiner Forest on -subgraph-free graphs remained tantalizingly open. In this paper, we make significant progress towards determining the complexity of Steiner Forest on -subgraph-free graphs. Our main results are four novel polynomial-time algorithms for different excluded graphs that are central to further understand its complexity. Along the way, we study the complexity of Steiner Forest for graphs with a small -deletion set, that is, a small set of vertices such that each component of has size at most . Using this parameter, we give two noteworthy algorithms that we later employ as subroutines. First, we prove Steiner Forest is FPT parameterized by when (i.e. the vertex cover number). Second, we prove Steiner Forest is polynomial-time solvable for graphs with a 2-deletion set of size at most 2. The latter result is tight, as the problem is NP-complete for graphs with a 3-deletion set of size 2.
Cite
@article{arxiv.2305.01613,
title = {Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem},
author = {Hans L. Bodlaender and Matthew Johnson and Barnaby Martin and Jelle J. Oostveen and Sukanya Pandey and Daniel Paulusma and Siani Smith and Erik Jan van Leeuwen},
journal= {arXiv preprint arXiv:2305.01613},
year = {2023}
}