The Algorithmic Complexity of Tree-Clique Width

TL;DR

This paper introduces tree-clique width, a new graph parameter extending tree-width to dense graphs, analyzes its computational complexity, and provides algorithms for specific graph classes.

Contribution

It defines tree-clique width, proves its NP-completeness, and offers algorithms for computing it on certain graph classes, expanding the applicability of width-based algorithms.

Findings

Tree-clique width is NP-complete.

No constant-factor approximation exists for any fixed c.

Algorithms are provided for cographs and permutation graphs.

Abstract

Tree-width has been proven to be a useful parameter to design fast and efficient algorithms for intractable problems. However, while tree-width is low on relatively sparse graphs can be arbitrary high on dense graphs. Therefore, we introduce tree-clique width, denoted by $tcl(G)$ for a graph $G$, a new width measure for tree decompositions. The main aim of such a parameter is to extend the algorithmic gains of tree-width on more structured and dense graphs. In this paper, we show that tree-clique width is NP-complete and that there is no constant-factor approximation algorithm for any constant value $c$. We also provide algorithms to compute tree-clique width for general graphs and for special graphs such as cographs and permutation graphs. We seek to understand further tree-clique width and its properties and to research whether it can be used as an alternative where tree-width fails.