Connected graph partitioning with aggregated and non-aggregated gap objective functions

This article deals with the problem of partitioning a graph into p connected components by optimizing some balancing objective functions related to the vertex weights. Objective functions based on the gap or range of the partition's components, that is, the difference between the maximum and minimum weight of a vertex in the component, have been already introduced in the literature. Here we introduce the notion of aggregated gap, defined as the sum of the differences between the weights of the vertices and the minimum weight of a vertex in the component. We study new connected p- partitioning problems whose objective is a function of the components' aggregated gap, and give NP-hardness results for these problems on general graphs. Mathematical programming formulations are proposed for these problems adopting flow-based constraints for modeling connectivity in a partition. Even if they are introduced for the new aggregated gap problems, such formulations are rather general and apply also to the classical non-aggregated gap problems. Extensive computational tests, both for aggregated and non-aggregated gap problems, are performed on a set of squared grids and randomly generated graphs with up to 120 vertices, and a number of components ranging from 2 to 9. In our experiments, we test several alternative formulations for our problems providing a comparative analysis of their performance.