A New Ant Colony Optimization Approach for the Degree-Constrained Minimum Spanning Tree Problem Using Prüfer and Blob Codes Tree Coding

This chapter describes a novel ACO algorithm for the degree-constrained minimum spanning tree (d-MST) problem. Instead of constructing the d-MST directly on the construction graph, ants construct the encoded d-MST. Two well-known tree codings are used: the Prüfer code, and the more recent Blob code (Picciotto, 1999). Both of these tree codings are bijective because they represent each spanning tree of the complete graph on |V| labelled vertices as a code of |V|-2 vertex labels. Each spanning tree corresponds to a unique code, and each code corresponds to a unique spanning tree. Under the proposed approach, ants will select graph vertices and place them into the Prüfer code or Blob code being constructed. The use of tree codings such as Prüfer code or Blob code makes it easier for the proposed ACO to solve another variant of the d-MST problem with both lower and upper bound constraints on each vertex (lu-dMST). A general lu-dMST problem formulation is given. This general lu-dMST problem formulation could be used to denote d-MST problem formulation also. Subsequently, Prüfer code and Blob code tree encoding and decoding are presented and then followed by the design of two ACO approaches using these tree codings to solve d-MST and lu-dMST problems. Next, results from these ACO approaches are compared on structured hard (SHRD) graph data set for both d-MST and ludMST problems, and important findings are reported.