Using Neighborhood Diversity to Solve Hard Problems

TL;DR

This paper explores neighborhood diversity as a graph parameter to develop efficient parameterized algorithms for NP-hard problems on dense graphs, extending prior work on vertex cover and tree-width.

Contribution

It introduces new algorithms parameterized by neighborhood diversity for p-Vertex-Disjoint Paths, Graph Motif, and Precoloring Extension problems.

Findings

Algorithms for these problems are fixed-parameter tractable with respect to neighborhood diversity.

Precoloring Extension remains hard on graphs of bounded neighborhood diversity.

Neighborhood diversity generalizes vertex cover to dense graphs, enabling new algorithmic approaches.

Abstract

Parameterized algorithms are a very useful tool for dealing with NP-hard problems on graphs. Yet, to properly utilize parameterized algorithms it is necessary to choose the right parameter based on the type of problem and properties of the target graph class. Tree-width is an example of a very successful graph parameter, however it cannot be used on dense graph classes and there also exist problems which are hard even on graphs of bounded tree-width. Such problems can be tackled by using vertex cover as a parameter, however this places severe restrictions on admissible graph classes. Michael Lampis has recently introduced neighborhood diversity, a new graph parameter which generalizes vertex cover to dense graphs. Among other results, he has shown that simple parameterized algorithms exist for a few problems on graphs of bounded neighborhood diversity. Our article further studies this area and provides new algorithms parameterized by neighborhood diversity for the p-Vertex-Disjoint Paths, Graph Motif and Precoloring Extension problems -- the latter two being hard even on graphs of bounded tree-width.