Dense Graph Partitioning on sparse and dense graphs

Cristina Bazgan; Katrin Casel; Pierre Cazals

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arXiv · 2107.13282

Dense Graph Partitioning on sparse and dense graphs

Cristina Bazgan·Katrin Casel·Pierre Cazals

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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 $t\geq 4$.

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BibTeX RIS

Cristina Bazgan, Katrin Casel, Pierre Cazals. 2021-07-28. Dense Graph Partitioning on sparse and dense graphs. https://arxiv.org/abs/2107.13282

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