A three-phase search approach for the quadratic minimum spanning tree problem

• QMSTP is a general model able to formulate a number of network design problems.
• We propose a three phase search heuristic (TPS) for this problem.
• TPS is assessed on 7 groups of 659 representative benchmarks of the literature.
• TPS finds improved best solutions for 33 challenging instances.
• TPS finds all the optimal solutions for the 29 instances transformed from the QAP.

Given an undirected graph with costs associated with each edge as well as each pair of edges, the quadratic minimum spanning tree problem (QMSTP) consists of determining a spanning tree of minimum cost. QMSTP is useful to model many real-life network design applications. We propose a three-phase search approach named TPS for solving QMSTP, which organizes the search process into three distinctive phases which are iterated: (1) a descent neighborhood search phase using two move operators to reach a local optimum from a given starting solution, (2) a local optima exploring phase to discover nearby local optima within a given regional area, and (3) a perturbation-based diversification phase to jump out of the current regional search area. TPS also introduces a pre-estimation criterion to significantly improve the efficiency of neighborhood evaluation, and develops a new swap-vertex neighborhood (as well as a swap-vertex based perturbation operator) which prove to be quite powerful for solving a series of special instances with particular structures. Computational experiments based on 7 sets of 659 popular benchmarks show that TPS produces highly competitive results compared to the best performing approaches in the literature. TPS discovers improved best known results (new upper bounds) for 33 open instances and matches the best known results for all the remaining instances. Critical elements and parameters of the TPS algorithm are analyzed to understand its behavior.