The parameterised complexity of computing the maximum modularity of a graph

TL;DR

This paper explores the computational complexity of finding the maximum modularity in graphs, revealing fixed-parameter tractability results for certain parameters and hardness results for others.

Contribution

It provides a detailed parameterised complexity analysis of the maximum modularity problem, identifying cases where it is efficiently solvable and cases where it is computationally hard.

Findings

FPT algorithms for vertex cover and bounded treewidth cases

Polynomial-time solvability for graphs with bounded treewidth or max leaf number

W[1]-hardness when parameterised by pathwidth and feedback vertex set size

Abstract

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP-complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.