Dense Graph Partitioning on sparse and dense graphs

TL;DR

This paper investigates the computational complexity and approximation algorithms for the Dense Graph Partition problem across various classes of sparse and dense graphs, revealing both hardness results and efficient solutions.

Contribution

It characterizes the complexity of Dense Graph Partition on specific graph classes and provides new polynomial-time algorithms and approximation schemes for certain dense graphs.

Findings

NP-hardness on dense bipartite and cubic graphs

Polynomial-time solvability on dense graphs with high minimum degree

A 4/3-approximation algorithm for cubic graphs

Abstract

We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on $n$ vertices, it is polynomial-time solvable on graphs with minimum degree $n-3$ and NP-hard on $(n-4)$-regular graphs. We prove that it is polynomial-time $4/3$-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree $n-t$ for any constant $t\geq 4$.