The Satisfactory Partition Problem

TL;DR

This paper investigates the parameterized complexity of the Satisfactory Partition problem, showing it is fixed-parameter tractable with respect to neighborhood diversity and clique-width, but W[1]-hard with respect to treewidth.

Contribution

It provides new fixed-parameter tractability results and hardness results for the problem based on different graph parameters.

Findings

FPT when parameterized by neighborhood diversity

Solvable in $O(n^{8 \text{cw}})$ time with respect to clique-width

W[1]-hard when parameterized by treewidth

Abstract

The Satisfactory Partition problem consists in deciding if the set of vertices of a given undirected graph can be partitioned into two nonempty parts such that each vertex has at least as many neighbours in its part as in the other part. This problem was introduced by Gerber and Kobler [European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors, but its parameterized complexity remains open until now. It is known that the Satisfactory Partition problem, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is FPT when parameterized by the neighbourhood diversity of the input graph, (2) it can be solved in $O(n^{8 {\tt cw}})$ where ${\tt cw}$ is the clique-width,(3) a generalized version of the problem is W[1]-hard when parameterized by the treewidth.