Polyhedral results, branch-and-cut and Lagrangian relaxation algorithms for the adjacent only quadratic minimum spanning tree problem

Given a complete and undirected graph G, the Adjacent Only Minimum Spanning Tree Problem (AQMSTP) consists of finding a spanning tree that minimizes a quadratic function of its adjacent edges. The strongest AQMSTP linear integer programming formulation in the literature works in an extended variable space, using exponentially many decision variables assigned to the stars of G. In this paper, we characterize two families of facet defining inequalities by investigating the projection of that formulation onto the space of the canonical linearization variables. On the algorithmic side, we introduce four new branch-and-bound (BB) algorithms. Three of them are branch-and-cut algorithms based on the