International Journal of Computer Applications (0975 – 8887) Volume 18– No.4, March 2011 17 Bounded-Diameter MST Instances with Hybridization of Multi-Objective EA Soma Saha Department of Computer Science & Engineering Indian Institute of Technology Kharagpur Kharagpur West Bengal India 721302 Rajeev Kumar Department of Computer Science & Engineering Indian Institute of Technology Kharagpur Kharagpur West Bengal India 721302 ABSTRACT The Bounded Diameter (a.k.a Diameter Constraint) Minimum Spanning Tree (BDMST/DCMST) is a well-known combinatorial optimization problem. In this paper, we recast a few well-known heuristics, which are evolved for BDMST problem to a Bi-Objective Minimum Spanning Tree (BOMST) problem and then obtain Pareto fronts. After examining Pareto fronts, it is concluded that none of the heuristics provides the superior solution across the complete range of the diameter. We have used a Multi-Objective Evolutionary Algorithm (MOEA) approach, Pareto Converging Genetic Algorithm (PCGA), to improve the Pareto front for BOMST, which in turn provides better solution for BDMST instances. We have considered edge-set encoding to represent MST and then applied recombination operators having strong heritability and mutation operators having negligible complexity to improve the solutions. Analysis of MOEA solutions confirms the improvement of Pareto front solutions across the complete range of the diameter over Pareto front solutions generated from individual heuristics. We have considered multi-island scheme using Inter-Island rank histogram and performed multiple run of the algorithm to avoid from trapping into local-optimal solution- set. General Terms Algorithm, Design, Experimentation. Keywords Combinatorial optimization, multi-objective optimization, heuristics, MST, BDMST problem, evolutionary algorithm, Pareto front, edge list encoding. 1. INTRODUCTION BDMST has many applications in real-world [2, 4, 22]; it is an NP-hard problem within diameter bound (D) ranges 4 ≤ D < |V| - 1 [9], where diameter bound (D) is a constraint, the maximum feasible longest path between two vertices of a connected, undirected, weighted graph G to generate feasible MSTs and V is the set of vertices of G. Well-known heuristics which are evolved to provide solutions to BDMST problem, are: e.g., one time tree construction (OTTC) [8], iterative refinement (IR) [8], randomized greedy heuristics (RGH) [21], random tree construction (RTC) [10], center based tree construction (CBTC) [10] and center-based recursive clustering (CBRC) [20]. Initially, algorithmic complexity directed the development of various heuristics. The complexity of OTTC, IR, CBTC and CBRC is O(n 3 ) [8, 10, 20]. Depending on initial vertex, generated spanning tree (ST) differs for OTTC, IR and CBTC heuristics; therefore, one needs to execute the heuristics with each of the vertices as initial vertex with the expectation to get better low-weight spanning tree and then select the best generated tree which makes the complexity O(n 4 ). RGH has complexity O(n 2 ) [21]. RGH generates ST randomly; thus, by increasing number of execution of the algorithm (say, no. of execution = n), the possibility to produce better solution is increased and the complexity of RGH raises to O(n 3 ). In this work, we have adapted well-known heuristics to formulate bi-objective MST problem and then obtained Pareto fronts [23, 24]. BDMST is a specialize instance of BOMST. Considering Pareto front solutions for each heuristic as initial population, we provide an MOEA approach to improve Pareto front which in turn improves the BDMST solution within a particular diameter constraint for that heuristic. We have considered various performance metrices to analyze the MOEA solution set which supports the improvement of quality of solutions. The paper is organized as follows, in Section 2, we include the basic definitions and outline problem formulation. We describe the overview of those heuristics which we deal in this work in Section 3. Next, Section 4 contains the evolutionary algorithm and the description of recombination and mutation operators. The results are contained in Section 5. We conclude with short discussion and future work in Section 6. 2. PRELIMINARY 2.1 Basic Definitions Definition 1 Multiobjective Optimization Problem (MOP). In an MOP, a number of objectives have to be minimized/ maximized along with constraints (optional) to achieve goal vectors which can be written as: Maximize/Minimize : F(X) = {f 1 (X), f 2 (X),…, f m (X)} subject to satisfaction of the constraints: C(X) = {c 1 (X), c 2 (X),…, c k (X)} ≤ (0,…,0) A set of objective values constitutes an objective space and the collection of decision variables forms a decision space. Definition 2 Pareto-optimal set. Without loss of generality, we assume Multi-Objective Spanning Tree (MOST) (includes BOMST) and BDMST are m-objective minimization problem. In an m-objective minimization problem, a vector of decision variables x  X' includes in Pareto-optimal (P) set as a Pareto-optimal point if another x* does not exist such that f i (x*) ≤ f j (x) for all i = 1, 2, 3,…,m and f i (x*) < f j (x) for atleast one j. Here, X' denotes the feasible region of the problem (i.e. where the constraints are satisfied).