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Structural Parameters for Steiner Orientation

Authors Tesshu Hanaka , Michael Lampis , Nikolaos Melissinos , Edouard Nemery , Hirotaka Ono , Manolis Vasilakis

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LIPIcs.ISAAC.2025.38.pdf
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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • ETH
  • Steiner Orientation
  • Treewidth

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Abstract

We consider the Steiner Orientation problem, where we are given as input a mixed graph G = (V,E,A) and a set of k demand pairs (s_i,t_i), i ∈ [k]. The goal is to orient the undirected edges of G in a way that the resulting directed graph has a directed path from s_i to t_i for all i ∈ [k]. We adopt the point of view of structural parameterized complexity and investigate the complexity of Steiner Orientation for standard measures, such as treewidth. Our results indicate that Steiner Orientation is a surprisingly hard problem from this point of view. In particular, our main contributions are the following:  
1) We show that Steiner Orientation is NP-complete on instances where the underlying graph has feedback vertex number 2, treewidth 2, pathwidth 3, and vertex integrity 6. 
2) We present an XP algorithm parameterized by vertex cover number vc of complexity n^O(vc²). Furthermore, we show that this running time is essentially optimal by proving that a running time of n^o(vc²) would refute the ETH. 
3) We consider parameterizations by the number of undirected or directed edges (|E| or |A|) and we observe that the trivial 2^|E| n^O(1)-time algorithm for the former parameter is optimal under the SETH. Complementing this, we show that the problem admits a 2^O(|A|) n^O(1)-time algorithm. 
In addition to the above, we consider the complexity of Steiner Orientation parameterized by tw+k (FPT), distance to clique (FPT), and vc+k (FPT with a polynomial kernel).

Cite As Get BibTex

Tesshu Hanaka, Michael Lampis, Nikolaos Melissinos, Edouard Nemery, Hirotaka Ono, and Manolis Vasilakis. Structural Parameters for Steiner Orientation. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.38

Author Details

Tesshu Hanaka
  • Kyushu University, Fukuoka, Japan
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Nikolaos Melissinos
  • Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Edouard Nemery
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Hirotaka Ono
  • Nagoya University, Japan
Manolis Vasilakis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France

Funding

  • Hanaka, Tesshu: Partially supported by JSPS KAKENHI Grant Numbers JP21K17707, JP22H00513, JP23H04388, JP25K03077, and JST, CRONOS, Japan Grant Number JPMJCS24K2.
  • Lampis, Michael: Partially supported by ANR project ANR-21-CE48-0022 (S-EX-AP-PE-AL).
  • Melissinos, Nikolaos: Partially supported by Charles University projects UNCE 24/SCI/008 and PRIMUS 24/SCI/012, and by the project 25-17221S of GAČR.
  • Ono, Hirotaka: Partially supported by JSPS KAKENHI Grant Numbers JP20H05967, JP22H00513, JP24K02898, JP25K03077, and JST, CRONOS, Japan Grant Number JPMJCS24K2.
  • Vasilakis, Manolis: Partially supported by ANR project ANR-21-CE48-0022 (S-EX-AP-PE-AL).

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