An FPT 2-Approximation for Tree-Cut Decomposition

TL;DR

This paper introduces the first parameterized 2-approximation algorithm for computing the tree-cut width of a graph, enabling more practical analysis of this parameter for complex graph problems.

Contribution

It provides the first constructive parameterized approximation algorithm for tree-cut width, improving the ability to compute or estimate this parameter efficiently.

Findings

Developed a 2-approximation algorithm with runtime $2^{O(w^2\log w)} \cdot n^2$.

Algorithm either confirms the tree-cut width exceeds w or finds a decomposition with width at most 2w.

No prior constructive algorithms existed for approximating tree-cut width.

Abstract

The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input $n$-vertex graph $G$ and an integer $w$, our algorithm either confirms that the tree-cut width of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$. Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.