Fixed-parameter tractability and hardness for Steiner rooted and locally connected orientations
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| Format: | Preprint |
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2025
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| _version_ | 1866912686680309760 |
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| author | Bérczi, Kristóf Hörsch, Florian Imolay, András Schwarcz, Tamás |
| author_facet | Bérczi, Kristóf Hörsch, Florian Imolay, András Schwarcz, Tamás |
| contents | Finding a Steiner strongly $k$-arc-connected orientation is particularly relevant in network design and reliability, as it guarantees robust communication between a designated set of critical nodes. Király and Lau (FOCS 2006) introduced a rooted variant, called the Steiner Rooted Orientation problem, where one is given an undirected graph on $n$ vertices, a root vertex, and a set of $t$ terminals. The goal is to find an orientation of the graph such that the resulting directed graph is Steiner rooted $k$-arc-connected. This problem generalizes several classical connectivity results in graph theory, such as those on edge-disjoint paths and spanning-tree packings. While the maximum $k$ for which a Steiner strongly $k$-arc-connected orientation exists can be determined in polynomial time via Nash-Williams' orientation theorem, its rooted counterpart is significantly harder: the problem is NP-hard when both $k$ and $t$ are part of the input. In this work, we provide a complete understanding of the problem with respect to these two parameters. In particular, we give an algorithm that solves the problem in time $f(k,t)\cdot n^{O(1)}$, establishing fixed-parameter tractability with respect to the number of terminals $t$ and the target connectivity $k$. We further show that the problem remains NP-hard if either $k$ or $t$ is treated as part of the input, meaning that our algorithm is essentially optimal from a parameterized perspective. Importantly, our results extend far beyond the Steiner setting: the same framework applies to the more general orientation problem with local connectivity requirements, establishing fixed-parameter tractability when parameterized by the total demand and thereby covering a wide range of arc-connectivity orientation problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02081 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fixed-parameter tractability and hardness for Steiner rooted and locally connected orientations Bérczi, Kristóf Hörsch, Florian Imolay, András Schwarcz, Tamás Discrete Mathematics Finding a Steiner strongly $k$-arc-connected orientation is particularly relevant in network design and reliability, as it guarantees robust communication between a designated set of critical nodes. Király and Lau (FOCS 2006) introduced a rooted variant, called the Steiner Rooted Orientation problem, where one is given an undirected graph on $n$ vertices, a root vertex, and a set of $t$ terminals. The goal is to find an orientation of the graph such that the resulting directed graph is Steiner rooted $k$-arc-connected. This problem generalizes several classical connectivity results in graph theory, such as those on edge-disjoint paths and spanning-tree packings. While the maximum $k$ for which a Steiner strongly $k$-arc-connected orientation exists can be determined in polynomial time via Nash-Williams' orientation theorem, its rooted counterpart is significantly harder: the problem is NP-hard when both $k$ and $t$ are part of the input. In this work, we provide a complete understanding of the problem with respect to these two parameters. In particular, we give an algorithm that solves the problem in time $f(k,t)\cdot n^{O(1)}$, establishing fixed-parameter tractability with respect to the number of terminals $t$ and the target connectivity $k$. We further show that the problem remains NP-hard if either $k$ or $t$ is treated as part of the input, meaning that our algorithm is essentially optimal from a parameterized perspective. Importantly, our results extend far beyond the Steiner setting: the same framework applies to the more general orientation problem with local connectivity requirements, establishing fixed-parameter tractability when parameterized by the total demand and thereby covering a wide range of arc-connectivity orientation problems. |
| title | Fixed-parameter tractability and hardness for Steiner rooted and locally connected orientations |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2511.02081 |