The Theory of Cyclic Transfers

This paper develops the theory of cyclic transfers, a neighborhood search strategy for a broad class of combinatorial problems. The problem class is characterized by a two-phase decision process: first, the optimal assignment of elements to clusters and, second, the optimal configuration of the elements within each cluster. Many important and difficult problems fall into this problem class, for example, facility location, multi-vehicle routing and scheduling, clustering, vertex covering, graph coloring, weighted matching and quadratic assignment problems. We characterize a local search neighborhood in terms of cyclic transfers of elements among clusters, define data structures useful in searching the neighborhood, and show that the neighborhood search problem is itself NPhard. We derive necessary and sufficient conditions for the existence of negative cost cyclic transfers in terms of negative cost cycles on an auxiliary graph. Finally, we examine algorithm implementation issues. 1C1-tr~rr~r-u ~·-a-^ C lxr-~---_r~~-· *r~il· Y --iC~ s mra rr~rs l·(·-r.i"eB-P-08 This paper develops and analyzes a generic type of local search algorithm * for a broad class of combinatorial problems. In the first section, we define rooted partitioning problems, the combinatorial problem class to which our research applies, and discuss representative examples. In Section 2, we motivate local search as a solution methodology and review the local search literature for this class of combinatorial problems. In the remaining sections, we develop the theory of cyclic transfers, the neighborhood search strategy that is the focus of this paper, and discuss implementation of cyclic transfer algorithms. Section 3 introduces cyclic transfers, characterizes a class of local neighborhoods in terms of cyclic transfers, shows that a solution locally optimal in a cyclic transfer neighborhood is not necessarily globally optimal, defines data structures useful in searching cyclic transfer neighborhoods, and investigates the computational complexity of the neighborhood search problem. Section 4 derives necessary and sufficient conditions for the existence of negative cost cyclic transfers in terms of negative cost cycles on an auxiliary graph. Section 5 examines algorithmic implementation issues. Finally, Section 6 presents conclusions. wF~ ~ We believe that the methodology developed in this paper will prove useful for solving a variety of problems of practical importance. Indeed, related work [Thompson, 1988] [Thompson and Psaraftis, 1989], has shown the efficacy of our methods for several classes of rooted partitioning problems that arise in vehicle routing and scheduling. 1. The Combinatorial Setting The generic class of combinatorial problems that we investigate in this paper we term rooted partitioning problems. Before defining this problem class, we introduce some notation. Let S be a set. An m-partition of S is a partition of S into m (possibly empty) subsets Il,I2,...,Im, i.e., the subsets (IJ:j-l to m) are pairwise disjoint, and their union is the set S. Now, let S and T be disjoint sets, ad let us refer to T as the root set, with T -m. A rooted partition of S u T is an m-partition of S T into subsets I,...,I m such that for each j 1 to m, I I n T 1. We refer to IJ as the j-th cluster, and we refer to the unique element in Ii n T as the root of the j-th cluster. We denote this element as ROOT(IJ).