Tight Bounds for Connectivity Problems Parameterized by Cutwidth

TL;DR

This paper establishes tight complexity bounds for several connectivity problems parameterized by cutwidth, providing new algorithms and matching lower bounds under SETH, and introduces a novel approach for coloring-like problems based on matrix rank.

Contribution

It offers the first tight bounds for multiple connectivity problems parameterized by cutwidth, extending existing methods and introducing a new matrix rank-based technique for coloring problems.

Findings

Matching lower bounds for odd cycle transversal and feedback vertex set.

Faster algorithms for connected vertex cover and connected dominating set.

A new matrix rank-based approach for solving coloring-like problems.

Abstract

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. 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 domintaing 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.~[STACS~2022] to solve what we call coloring-like problem. Such problems are defined by a symmetric matrix $M$ over $\mathbb{F}_2$ 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 $\mathbb{F}_p$. We apply this result to get our upper bounds for connected vertex cover and connected dominating set. The upper bounds for odd cycle transversal and feedback vertex set use a subdivision trick to get below the bounds that matrix rank would yield.