Fast exact algorithms for some connectivity problems parametrized by clique-width

TL;DR

This paper introduces fast exact algorithms with running times of 2^{O(k)}·n for various connectivity problems on graphs, leveraging clique-width expressions to improve efficiency over previous methods.

Contribution

It presents the first algorithms with single-exponential dependence on clique-width for several connectivity problems, improving upon prior exponential-logarithmic bounds.

Findings

Algorithms run in 2^{O(k)}·n time for multiple problems.

Significant improvement over previous exponential-logarithmic time algorithms.

Applicable to problems like Steiner Tree, Dominating Set, Vertex Cover, and Feedback Vertex Set.

Abstract

Given a clique-width $k$-expression of a graph $G$, we provide $2^{O(k)}\cdot n$ time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex Cover. We also propose a $2^{O(k)}\cdot n$ time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either $2^{O(k\cdot \log(k))}\cdot n^{O(1)}$ or worse.