On the generalized minimum spanning tree problem

This paper considers the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose nodes are partitioned into mutually exclusive and exhaustive node sets, The GMSTP is then to find a minimum‐cost tree which includes exactly one node from each node set. Here, we show that the GMSTP is NP ‐hard and that unless P = NP no polynomial‐time heuristic algorithm with a finite worst‐case performance ratio can exist for the GMSTP. We present various integer programming formulations for the problem and compare their linear programming relaxations. Based on the tightest formulation among the ones proposed, a dual‐based solution procedure is developed and shown to be efficient from computing experiments.