Parameterized Algorithms for Modular-Width

TL;DR

This paper introduces the modular-width parameter for graphs, enabling fixed-parameter tractable algorithms for problems like Coloring and Hamiltonian paths, which are hard under other parameters like clique-width.

Contribution

It proposes modular-width as a versatile graph parameter that generalizes simpler notions and supports FPT algorithms for problems previously W-hard under clique-width.

Findings

FPT algorithms for Coloring and Partitioning into paths using modular-width

Modular-width generalizes several graph parameters for dense graphs

Shows modular-width avoids the computational hardness of clique-width

Abstract

It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty. The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILPs and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path and Hamiltonian cycle), which are W-hard for both clique-width and its recently introduced restriction, shrub-depth. We thus argue that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.