2 research outputs found
A tight Monte-Carlo algorithm for Steiner Tree parameterized by clique-width
Recently, Hegerfeld and Kratsch [ESA 2023] obtained the first tight
algorithmic results for hard connectivity problems parameterized by
clique-width. Concretely, they gave one-sided error Monte-Carlo algorithms that
given a -clique-expression solve Connected Vertex Cover in time
and Connected Dominating Set in time . Moreover,
under the Strong Exponential-Time Hypothesis (SETH) these results were showed
to be tight. However, they leave open several important benchmark problems,
whose complexity relative to treewidth had been settled by Cygan et al. [SODA
2011 & TALG 2018]. Among which is the Steiner Tree problem. As a key
obstruction they point out the exponential gap between the rank of certain
compatibility matrices, which is often used for algorithms, and the largest
triangular submatrix therein, which is essential for current lower bound
methods. Concretely, for Steiner Tree the -rank is , while no
triangular submatrix larger than was known. This yields time
, while the obtainable impossibility of time
under SETH was already known relative to pathwidth.
We close this gap by showing that Steiner Tree can be solved in time
given a -clique-expression. Hence, for all parameters between
cutwidth and clique-width it has the same tight complexity. We first show that
there is a ``representative submatrix'' of GF(2)-rank (ruling out larger
triangular submatrices). At first glance, this only allows to count (modulo 2)
the number of representations of valid solutions, but not the number of
solutions (even if a unique solution exists). We show how to overcome this
problem by isolating a unique representative of a unique solution, if one
exists. We believe that our approach will be instrumental for settling further
open problems in this research program
Tight Bounds for Connectivity Problems Parameterized by Cutwidth
In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. [Bas A. M. van Geffen et al., 2020] posed this question for Odd Cycle Transversal and Feedback Vertex Set. We answer it for these two and four further problems, namely Connected Vertex Cover, Connected Dominating Set, Steiner Tree, and Connected Odd Cycle Transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al. [Carla Groenland et al., 2022] to solve what we call coloring-like problems. Such problems are defined by a symmetric matrix M over ?? indexed by a set of colors. The goal is to count the number (modulo some prime p) of colorings of a graph such that M has a 1-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if M has small rank over ?_p. We apply this result to get our upper bounds for CVC and CDS. The upper bounds for OCT and FVS use a subdivision trick to get below the bounds that matrix rank would yield