Tight algorithms for connectivity problems parameterized by clique-width

TL;DR

This paper develops optimal algorithms for connectivity problems parameterized by clique-width, specifically for Connected Vertex Cover and Connected Dominating Set, establishing tight bounds under SETH.

Contribution

It introduces the first tight algorithms for these problems parameterized by clique-width, matching lower bounds and advancing understanding of dense graph structures.

Findings

Algorithms run in 6^k n^O(1) for Connected Vertex Cover

Algorithms run in 5^k n^O(1) for Connected Dominating Set

Results are tight under SETH

Abstract

The complexity of problems involving global constraints is usually much more difficult to understand than the complexity of problems only involving local constraints. A natural form of global constraints are connectivity constraints. We study connectivity problems from a fine-grained parameterized perspective. In a breakthrough, Cygan et al. (TALG 2022) first obtained algorithms with single-exponential running time c^{tw} n^O(1) for connectivity problems parameterized by treewidth by introducing the cut-and-count-technique. Furthermore, the obtained bases c were shown to be optimal under the Strong Exponential-Time Hypothesis (SETH). However, since only sparse graphs may admit small treewidth, we lack knowledge of the fine-grained complexity of connectivity problems with respect to dense structure. The most popular graph parameter to measure dense structure is arguably clique-width, which intuitively measures how easily a graph can be constructed by repeatedly adding bicliques. Bergougnoux and Kant\'e (TCS 2019) have shown, using the rank-based approach, that also parameterized by clique-width many connectivity problems admit single-exponential algorithms. Unfortunately, the obtained running times are far from optimal under SETH. We show how to obtain optimal running times parameterized by clique-width for two benchmark connectivity problems, namely Connected Vertex Cover and Connected Dominating Set. These are the first tight results for connectivity problems with respect to clique-width and these results are obtained by developing new algorithms based on the cut-and-count-technique and novel lower bound constructions. Precisely, we show that there exist one-sided error Monte-Carlo algorithms that given a k-clique-expression solve Connected Vertex Cover in time 6^k n^O(1), and Connected Dominating Set in time 5^k n^O(1). Both results are shown to be tight under SETH.