Abstract
We study Steiner Forest on H-subgraph-free graphs, that
is, graphs that do not contain some fixed graph H as a (not necessarily
induced) subgraph. We are motivated by a recent framework that completely characterizes the complexity of many problems on H-subgraphfree 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 H-subgraph-free graphs remained tantalizingly open. We make significant progress on this open problem: our main
results are four novel polynomial-time algorithms for different excluded
graphs H that are central to further understand its complexity. Along
the way, we study the complexity of Steiner Forest for graphs with
a small c-deletion set, that is, a small set X of vertices such that each
component of G−X has size at most c. Using this parameter, we give two
algorithms that we later employ as subroutines. First, we present a significantly faster parameterized algorithm for Steiner Forest parameterized by |X| when c = 1 (i.e. the vertex cover number), which by a recent
result is best possible under ETH [Feldmann and Lampis, arXiv 2024].
Second, we prove that 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.
is, graphs that do not contain some fixed graph H as a (not necessarily
induced) subgraph. We are motivated by a recent framework that completely characterizes the complexity of many problems on H-subgraphfree 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 H-subgraph-free graphs remained tantalizingly open. We make significant progress on this open problem: our main
results are four novel polynomial-time algorithms for different excluded
graphs H that are central to further understand its complexity. Along
the way, we study the complexity of Steiner Forest for graphs with
a small c-deletion set, that is, a small set X of vertices such that each
component of G−X has size at most c. Using this parameter, we give two
algorithms that we later employ as subroutines. First, we present a significantly faster parameterized algorithm for Steiner Forest parameterized by |X| when c = 1 (i.e. the vertex cover number), which by a recent
result is best possible under ETH [Feldmann and Lampis, arXiv 2024].
Second, we prove that 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.
Original language | English |
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Title of host publication | Combinatorial Algorithms - 35th International Workshop, IWOCA 2024, Ischia, Italy, July 1-3, 2024, Proceedings |
Publisher | Springer |
Pages | 206-217 |
ISBN (Electronic) | 978-3-031-63021-7 |
ISBN (Print) | 978-3-031-63020-0 |
DOIs | |
Publication status | Published - 22 Jun 2024 |
Publication series
Name | Lecture Notes in Computer Science |
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Publisher | Springer |
Volume | 14764 |