Exact solution of the 2-dimensional grid arrangement problem

Abstract

Given an undirected graph G=(V,E), we consider injective mappings of its vertices to the k-dimensional Cartesian integer grid. Among such embeddings we are interested in those that minimize the sum of the resulting edge lengths, where the length of an edge is defined by the L1-metric. The case k=1 is the well-known Minimum Linear Arrangement Problem. We prove that the general problem is NP-hard for any fixed grid dimension. Our computational study focuses on the case k=2. We present as a first exact optimization algorithm a branch-and-cut scheme for sparse graphs. Several classes of valid inequalities are introduced and analyzed for facet defining properties of two corresponding polyhedra. Finally, computational results from a successful real-world application of the problem at the German Cancer Research Center (DKFZ) are presented.