Chapter 4 Acyclic Partitioning Problem
This chapter addresses the problem of partitioning the vertex set of a given directed, arc- and vertex-weighted graph into disjoint subsets (i.e., clusters) as considered in Nossack and Pesch (2012a) and Briskorn et al. (2012). Clus- ters are to be determined such that the sum of the vertex weights within the clusters satisﬁes an upper bound and the sum of the arc 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 cy- cle. This problem is known in the literature as acyclic partitioning problem and is proven to be NP-hard in the strong sense (refer, e.g., to Garey and Johnson (1979)). Real-life applications arise, e.g., at rail-rail transshipment yards and in VLSI (Very Large Scale Integration) design. We propose sev- eral integer programming formulations for the acyclic partitioning problem and suggest two solution approaches, a branch-and-bound framework that integrates constraint propagation and a branch-and-price algorithm. Com- putational results are reported to verify the strength of our proposals. 4.1 Problem Description Graph partitioning problems are, in general, concerned with the partition- ing of the vertex set of an undirected or directed graph into disjoint subsets such that the sum of the edge weights within the clusters is maximized (or equivalently, the sum of the edge weights between diﬀerent clusters is min- 80 Chapter 4. Acyclic Partitioning Problem (a) Digraph (b) Acyclic Partition ! ! ! ! (c) Partition Fig. 4.1: Partition and Acyclic Partition imized). Most graph...
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