Using Genetic Algorithms to solve the Minimum Labeling Spanning Tree Problem

Genetic Algorithms (GAs) have shown themselves to be very powerful tools for a wide variety of combinatorial optimization problems. Through this project I hope to implement a GA to solve the Minimum Labeling Spanning Tree (MLST) problem (a combinatorial optimization problem). If time permits, I may attempt to modify the code to solve another combinatorial optimization problem. Additionally, I will develop a parallel implementation of the GA, which will involve designing and testing various inter-processor communication schemes. The eventuating parallel GA will be tested on a database of problems, comparing results and running time with other serial heuristics proposed in the literature. Finally, the parallel heuristic will be analyzed to determine how performance scales with the number of processors. 1 Background and Introduction 1.1 Problem: The Minimum Labeled Spanning Tree The Minimum Labeling Spanning Tree (MLST) was first proposed in 1996 by Chang and Leu [4] as a variant on the Minimum Weight Spanning Tree problem. In it we are given a connected graph G (composed of edges, E, and vertices, V ). Each edge is given one label (not necessarily unique) from the set L. We denote |E| = e, |V | = v and |L| = l. One such graph is shown in Figure 1. A sub-graph is generated by only using the edges from a subset C ⊂ L. The aim of the problem is to find the smallest possible set of labels which will generate a connected subgraph. More than one global minimum (equally small sets) may exist, although we are satisfied if we identify one. Real world applications include the design of telecommunication [12] and computer networks [15] This problem has been shown to be NP-Complete [4], and we therefore must use a heuristic to obtain near-optimal results in a reasonable amount of time (guaranteeing this is the true optimum solution will take unreasonably long). In this paper a solution is a set of labels, and we will call a set ’feasible’ if the sub-graph generated by the set of labels is connected. 1.2 Existing (non-GA) Heuristics Several heuristics have been proposed to solve the MLST problem. Appendix B contains pseudo-code for one such heuristic, the Maximum Vertex Covering Algorithm (MVCA) by Chang and Leu [4]. Other heuristics that have been used include Simulated Annealing, Tabu Searches, Pilot Methods, Variable Neighborhood Searches and a Greedy Randomized Adaptive Search [3,6]. I hope to be able to compare my algorithm with several of these heuristics.