Tight algorithms for connectivity problems parameterized by clique-width
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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 Kanté (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.
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