New Parameterized and Exact Exponential Time Algorithms for Strongly Connected Steiner Subgraph

2026-04-28

AI Summary

This study addresses the Strongly Connected Steiner Subgraph (SCSS) problem in directed graphs, which seeks the smallest strongly connected subgraph containing a given set of terminals. The work presents the first parameterized algorithm for SCSS with running time $17^{tw} \cdot n^{O(1)}$, where $tw$ denotes the treewidth, significantly improving upon prior approaches. It also refines the best-known exact algorithm for general directed graphs to run in $2^n \cdot n^{O(1)}$ time. Furthermore, the paper establishes that SCSS does not admit a polynomial kernel when parameterized by vertex cover number, unless NP ⊆ coNP/poly. By integrating tree decompositions, parameterized complexity analysis, exact exponential-time algorithm design, and kernelization lower bounds, this research advances both the practical solvability and theoretical understanding of SCSS across multiple parameterizations.

Abstract

The Strongly Connected Steiner Subgraph (SCSS) problem is a well-studied network design problem that asks for a minimum subgraph that strongly connects a given set of terminals. In this paper, we present several new algorithmic and complexity results for SCSS. As our main result, we show that SCSS can be solved in time $17^{\mathrm{tw}} n^{O(1)}$ on directed graphs with $n$ vertices when a tree decomposition of the underlying graph of width $\mathrm{tw}$ is provided. This improves over a natural $\mathrm{tw}^{O(\mathrm{tw})}n^{O(1)}$ time algorithm, and is the first algorithm with this kind of running time for a problem involving strong connectivity. Second, we give an exact exponential-time algorithm that solves SCSS in $2^n n^{O(1)}$ time, improving the known bounds for general directed graphs. Finally, we investigate kernelization with respect to vertex cover. We prove that SCSS does not admit a polynomial kernel when parameterized by the size of a vertex cover, unless the polynomial hierarchy collapses. In contrast, we show that the closely related Strongly Connected Spanning Subgraph problem does admit a polynomial kernel under the same parameterization.

Problem

• Strongly Connected Steiner Subgraph
• SCSS
• network design
• parameterized complexity
• exact exponential time

Innovation

• Strongly Connected Steiner Subgraph
• fixed-parameter tractability
• tree decomposition
• exact exponential algorithm
• polynomial kernel