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-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 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.
J.J. Oostveen is supported by the NWO grant OCENW.KLEIN.114 (PACAN).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alekseev, V.E., Korobitsyn, D.V.: Complexity of some problems on hereditary graph classes. Diskret. Mat. 4, 34–40 (1992)
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)
Barefoot, C.A., Entringer, R., Swart, H.: Vulnerability in graphs - a comparative survey. J. Comb. Math. Comb. Comput. 1, 13–22 (1987)
Bateni, M., Hajiaghayi, M.T., Marx, D.: Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth. J. ACM 58, 21:1–21:37 (2011)
Bodlaender, H.L., Brettell, N., Johnson, M., Paesani, G., Paulusma, D., van Leeuwen, E.J.: Steiner trees for hereditary graph classes: a treewidth perspective. Theoret. Comput. Sci. 867, 30–39 (2021)
Bodlaender, H.L., et al.: Subgraph Isomorphism on graph classes that exclude a substructure. Algorithmica 82, 3566–3587 (2020)
Bulteau, L., Dabrowski, K.K., Köhler, N., Ordyniak, S., Paulusma, D.: An algorithmic framework for locally constrained homomorphisms. In: Bekos, M.A., Kaufmann, M. (eds.) WG 2022. LNCS, vol. 13453, pp. 114–128. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15914-5_9
Drange, P.G., Dregi, M.S., van ’t Hof, P.: On the computational complexity of vertex integrity and component order connectivity. Algorithmica 76, 1181–1202 (2016)
Dvorák, P., Eiben, E., Ganian, R., Knop, D., Ordyniak, S.: Solving integer linear programs with a small number of global variables and constraints. In: Proceedings of IJCAI 2017, pp. 607–613 (2017)
Feldmann, A.E., Lampis, M.: Parameterized algorithms for steiner forest in bounded width graphs. In: Proceedings of ICALP 2024. LIPIcs (2024, to appear)
Fujita, S., Furuya, M.: Safe number and integrity of graphs. Discret. Appl. Math. 247, 398–406 (2018)
Fujita, S., MacGillivray, G., Sakuma, T.: Safe set problem on graphs. Discret. Appl. Math. 215, 106–111 (2016)
Gima, T., Hanaka, T., Kiyomi, M., Kobayashi, Y., Otachi, Y.: Exploring the gap between treedepth and vertex cover through vertex integrity. Theoret. Comput. Sci. 918, 60–76 (2022)
Golovach, P.A., Paulusma, D.: List coloring in the absence of two subgraphs. Discret. Appl. Math. 166, 123–130 (2014)
Johnson, M., Martin, B., Oostveen, J.J., Pandey, S., Paulusma, D., Smith, S., van Leeuwen, E.J.: Complexity framework for forbidden subgraphs I: the framework. CoRR abs/2211.12887 (2022)
Johnson, M., Martin, B., Pandey, S., Paulusma, D., Smith, S., van Leeuwen, E.J.: Edge multiway cut and node multiway cut are hard for planar subcubic graphs. In: Proceedings of SWAT 2024. LIPIcs, vol. 294, pp. 29:1–29:17 (2024)
Johnson, M., Martin, B., Pandey, S., Paulusma, D., Smith, S., van Leeuwen, E.J.: Complexity framework for forbidden subgraphs III: when problems are polynomial on subcubic graphs. In: Proceedings of MFCS 2023, LIPIcs, vol. 272, pp. 57:1–57:15 (2023)
Kamiński, M.: Max-cut and containment relations in graphs. Theoret. Comput. Sci. 438, 89–95 (2012)
Martin, B., Pandey, S., Paulusma, D., Siggers, M., Smith, S., van Leeuwen, E.J.: Complexity framework for forbidden subgraphs II: when hardness is not preserved under edge subdivision. CoRR abs/2211.14214 (2022)
Acknowledgments
We thank Daniel Lokshtanov for pointing out a possible relationship between Steiner Forest and CSP, as discussed in Sect. 5. We also thank the anonymous reviewers of earlier versions of this paper for their helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bodlaender, H.L. et al. (2024). Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem. In: Rescigno, A.A., Vaccaro, U. (eds) Combinatorial Algorithms. IWOCA 2024. Lecture Notes in Computer Science, vol 14764. Springer, Cham. https://doi.org/10.1007/978-3-031-63021-7_16
Download citation
DOI: https://doi.org/10.1007/978-3-031-63021-7_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-63020-0
Online ISBN: 978-3-031-63021-7
eBook Packages: Computer ScienceComputer Science (R0)