Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth

TL;DR

This paper introduces a hybrid framework that extends treewidth-based dynamic programming to graphs with complex structures, enabling fixed-parameter algorithms for problems like Chromatic Number and Max-Cut.

Contribution

It generalizes existing algorithms by incorporating a refined treewidth concept that handles hybrid graph structures with additional parameters.

Findings

Developed a framework for hybrid graph structures.

Achieved fixed-parameter algorithms for key problems.

Combined treewidth and rank-width for improved algorithms.

Abstract

We develop a framework for applying treewidth-based dynamic programming on graphs with "hybrid structure", i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.