Abstract— In this paper, a new hybrid genetic algorithm – known as HGA – is proposed for solving the Bounded Diameter Minimum Spanning Tree (BDMST) problem. We experiment with HGA on two sets of benchmark problem instances, both Euclidean and Non-Euclidean. On the Euclidean problem instances, HGA is shown to outperform the previous best two Genetic Algorithms (GAs) reported in the BDMST literature, while on the Non-Euclidean problem instance, HGA performs comparably with these two GAs. I. INTRODUCTION he bounded diameter minimum spanning tree (BDMST) problem is a combinatorial optimization problem that arises in many applications such as design of wire-based communication networks under quality of service requirements, in ad-hoc wireless networks [3], in data compression and in distributed mutual exclusion algorithms [2, 6]. A more comprehensive discussion of the real-world applications of BDMST was given in Abdalla’s seminal dissertation [10]. Before the BDMST problem can be formally stated, we need to define some concepts relating to tree diameter and center. Given a tree T, the maximal eccentricity of vertex v is the length (measured in the number of edges) of the longest path from v to other vertices. The diameter of a tree T, denoted as diam(T), is the maximal eccentricity over all nodes in T (i.e the length of maximal path between two arbitrary vertices in T). Suppose that a diameter of a tree is defined by the path v 1 , v 2 ,…, v [k/2] , v [k/2]+1 , …, v k . If k is even then v [k/2] is called a center of the tree. If k is odd then v [k/2] and v [k/2]+1 are centers of the tree. In that case, the edge (v [k/2] , v [k/2]+1 ) is called a center edge. Let G = (V, E) be a connected undirected graph with positive edge weights w(e). The BDMST problem can be formulated as follows: among spanning trees of G whose diameters do not exceed a given upper bound D≥2, find the spanning tree with the minimal cost (sum of the weights on edges of the trees). As in almost all studies of the BDMST problem, and without lost of generality, we will assume that G is a complete graph. That is, we can formulate the problem as: Huynh Thi Thanh Binh is with the Department of Information Technology, Hanoi University of Technology, 1 st Dai Co Viet St., Hanoi, Vietnam. E-mail: BinhHT@it-hut.edu.vn Nguyen Xuan Hoai and Bob McKay are with the school of Computer Science and Engineering, Seoul National University, Korea. E-mails: nxhoai@cse.snu.ac.kr; rim@cse.snu.ac.kr Find a spanning tree T = (V, E ) of G that minimizes ( ) () eT WT we ∈ = ∑ subject to diam(T) ≤ D. This problem is known to be NP-hard for 4 ≤ D ≤ |V|-1 [1]. Moreover, the BDMST problem has been shown to be also approximate-hard, in that there is no polynomial time algorithm which could guarantee to find a solution that has a cost within log(|V|) of the optimum [22]. Therefore, heuristic and meta-heuristic techniques are currently the only method for improving the solution quality in solving the BDMST problem, especially when n is large. In this paper, we introduce a new hybrid genetic algorithm (HGA) for solving BDMST problems. The new genetic algorithm use a multi-population, where each population is initialized with a different well known heuristic. The individuals in each population will subsequently compete for positions in a selection population, using a simulated annealing mechanism based on proportionate selection; in the selection population, they will combine and evolve toward the optimum. We compare our results with two other genetic algorithms on the same problem, namely, the genetic algorithm in [12] of Raidl and Julstrom (called RJ-ESEA in this paper), and the genetic algorithm of Alok and Gupta in [21] (called PEA-I). The paper is organized as follows. In the next section (section 2), we briefly overview work done in solving BDMST problems, highlighting the motivations for our work. Section 3 contains the description of our new hybrid genetic algorithm for the BDMST problem. The details of our experiments are given in section 4, while the computational and comparative results are given and discussed in section 5 of the paper. The paper concludes with section 6, where we also describe some possible future extensions of this work. II. PREVIOUS WORK ON THE BDMST PROBLEM Techniques for solving the BDMST problem may be classified into two categories: exact methods and inexact (heuristic) methods. Exact approaches for solving the BDMST problem are based on mixed linear integer programming [5, 15]. More recently, Gruber and Raidl suggested a branch and cut algorithm based on compact 0-1 integer linear programming [16]. However, being deterministic and exhaustive in nature, these approaches A New Hybrid Genetic Algorithm for Solving the Bounded Diameter Minimum Spanning Tree Problem Huynh Thi Thanh Binh, Nguyen Xuan Hoai, and R.I (Bob) McKay T