Fixed-parameter tractability and hardness for Steiner rooted and locally connected orientations

TL;DR

This work studies Steiner Rooted $k$-arc-connected orientations, showing that the problem is fixed-parameter tractable when parameterized by the target connectivity $k$ and the number of terminals $t$, with running time $f(k,t) abla n^{O(1)}$. The authors develop a framework combining fixed-topological-minor testing, extremal results like Erdős–Pósa for directed cycles, and Ramsey-type arguments to bound and enumerate a finite set of minimal 3-regular templates. They prove that the same framework yields tractability for the more general $R$-Orientation with total demand $oldsymbol\alpha$, while also establishing NP-hardness for fixed $k$ (with $t$ part of the input) and for fixed $t\ge4$ (with $k$ part of the input), thereby giving a near-complete complexity landscape. The results extend to local connectivity variants, providing a broad parameterized understanding of orientation problems with higher connectivity demands and offering concrete tools for algorithmic design in network reliability and related domains.

Abstract

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.

Figures (4)

• Figure 1: An illustration of the proof of \ref{['lem:deg3']} with $k=2$ and $S=\{s\}$.
• Figure 2: An illustration of the proof of \ref{['serdgh']}. $C=\{x_1, x_2, x_3\}$ is a clause, and the orientation corresponds to a truth assignment $\phi$ with $\phi(x_1)=\texttt{False}$ and $\phi(x_2)=\phi(x_3)=\texttt{True}$.
• Figure 3: An illustration of the proof of \ref{['etxdrcdzfvzugbu']}. The set of clauses consists of $\mathcal{C}_1=\{(x_1\vee x_2), (x_3\vee \widebar{x_4}), (\widebar{x_5}\vee x_6)\}$, $\mathcal{C}_2=\{(x_1\vee \widebar{x_3}, ), (\widebar{x_2}\vee x_4), (\widebar{x_5}\vee \widebar{x_6}))\}$, and $\mathcal{C}_3=\{(\widebar{x_2}\vee \widebar{x_4}), (x_5 \vee \widebar{x_6})\}$. The orientation corresponds to the truth assignment $\phi$ with $\phi(x_1)=\phi(x_4)=\texttt{True}$, $\phi(x_2)=\phi(x_3)=\phi(x_5)=\phi(x_6)=\texttt{False}$.
• Figure 4: An illustration of the proof of \ref{['tfix']}.

Theorems & Definitions (87)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof