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

TL;DR

This paper introduces new parameterized and exponential algorithms for the Strongly Connected Steiner Subgraph problem, improving existing bounds and analyzing kernelization complexity.

Contribution

It presents the first fixed-parameter algorithm with a $17^{tw}$ running time and an improved exponential algorithm for SCSS, along with kernelization complexity results.

Findings

SCSS can be solved in $17^{tw} n^{O(1)}$ time with a tree decomposition.

An exact exponential-time algorithm for SCSS runs in $2^n n^{O(1)}$ time.

SCSS does not admit a polynomial kernel parameterized by vertex cover size.

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.