The quadratic minimum spanning tree problem: A lower bounding procedure and an efficient search algorithm

In this work we address the Quadratic Minimum Spanning Tree Problem (QMSTP) known to be NP-hard. Given a complete graph, the QMSTP consists of determining a minimum spanning tree (MST) considering interaction costs between pairs of edges to be modelled. A Lagrangian relaxation procedure is devised and an efficient local search algorithm with tabu thresholding is developed. Computational experiments are reported on randomly generated test instances. The local search heuristic yields at least as well as the results obtained with a previous Genetic Algorithm and the Lagrangian relaxation procedure gives tightest lower bound values for all instances.