Dense Graph Partitioning on sparse and dense graphs [article]

Cristina Bazgan, Katrin Casel, Pierre Cazals
2022 arXiv   pre-print
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
more » ... y 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≥ 4.
arXiv:2107.13282v3 fatcat:l2eaob2omzfizh3sdvkykrj5yy