Mathematical Formulations for the Acyclic Partitioning Problem

This paper addresses the problem of partitioning the vertex set of a given directed, edge- and vertex-weighted graph into disjoint subsets (i.e., clusters). Clusters are to be determined such that the sum of the vertex weights within the clusters satisfies an upper bound and the sum of the edge weights within the clusters is maximized. Additionally, the digraph is enforced to partition into a directed, acyclic graph, i.e., a digraph that contains no directed cycle. This problem is known in the literature as acyclic partitioning problem and is proven to be NP-hard in the strong sense. Real-life applications arise, e.g., at rail-rail transshipment yards and in Very Large Scale Integration (VLSI) design. We propose two model formulations for the acyclic partitioning problem, a compact and an augmented set partitioning model.