A Unifying Framework for Characterizing and Computing Width Measures

TL;DR

This paper introduces F-branchwidth, a unifying framework that captures various width measures like treewidth, clique-width, and mim-width, enabling efficient algorithms for their computation and approximation.

Contribution

The work defines F-branchwidth as a generic class of parameters that unify multiple width measures and develops algorithms to compute them efficiently.

Findings

F-branchwidth captures multiple width measures including mim-width, treewidth, and clique-width.

Only a few F-branchwidth parameters are asymptotically distinct.

Algorithms are provided for computing F-branchwidth under various structural parameters.

Abstract

Algorithms for computing or approximating optimal decompositions for decompositional parameters such as treewidth or clique-width have so far traditionally been tailored to specific width parameters. Moreover, for mim-width, no efficient algorithms for computing good decompositions were known, even under highly restrictive parameterizations. In this work we identify F-branchwidth as a class of generic decompositional parameters that can capture mim-width, treewidth, clique-width as well as other measures. We show that while there is an infinite number of F-branchwidth parameters, only a handful of these are asymptotically distinct. We then develop fixed-parameter and kernelization algorithms (under several structural parameterizations) that can compute every possible F-branchwidth, providing a unifying framework that can efficiently obtain near-optimal tree-decompositions, k-expressions, as well as optimal mim-width decompositions.