Exact and Heuristic Approaches for Solving the Bounded Diameter Minimum Spanning Tree Problem

The bounded diameter minimum spanning tree (BDMST) problem is a combinatorial optimization problem appearing in applications such as wire-based communication network design when quality of service is of major concern and, for example, a signal between any two nodes in the network should not pass more than a fixed number of routers. It also arises in ad-hoc wireless networks and in the areas of data compression and distributed mutual exclusion algorithms. Given an undirected, weighted, and connected graph G = (V,E) with a node set V and an edge set E the goal is to identify a tree structure of minimum costs connecting all nodes of this network where the number of edges on each path linking any pair of nodes is limited by a maximum diameter D. This problem is known to be NP-hard for a diameter bound of 4 ≤ D < |V | − 1. There exist a great variety of different approaches to solve this problem. Beside models based on Miller-Tucker-Zemlin inequalities especially multi-commodity hopindexed flow formulations have proven to be very successful in exactly solving the BDMST problem. However, to obtain tight linear programming (LP) relaxation bounds these models require a huge number of variables. Due to the complexity of the problem exact approaches are limited to relatively small instances with clearly less than 100 nodes when considering complete graphs. Therefore, heuristics have been developed to solve instances with up to 1000 and more nodes. Fast and simple greedy construction heuristics are primarily based on Prim’s minimum spanning tree algorithm, but in particular on Euclidean instances this greedy behavior misleads these heuristics since in this case in general also long edges are part of a good BDMST. For higher quality solutions metaheuristics, especially evolutionary algorithms, have been proposed.