Cutwidth: obstructions and algorithmic aspects

TL;DR

This paper investigates the minimal obstructions for graphs with bounded cutwidth, providing size bounds and a new fixed-parameter algorithm that is simpler and more self-contained than previous methods.

Contribution

It establishes size bounds for minimal immersion obstructions for cutwidth and introduces a new fixed-parameter algorithm for computing cutwidth that is simpler and self-contained.

Findings

Minimal immersion obstructions have size at most exponential in k^3 log k.

New fixed-parameter algorithm runs in 2^{O(k^2 log k)}·n time.

Algorithm is simpler and more self-contained than previous fastest algorithms.

Abstract

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.