The minimum area spanning tree problem: Formulations, Benders decomposition and branch-and-cut algorithms

Abstract

The Minimum Area Spanning Tree Problem (MASTP) is defined in terms of a complete undirected graph G, where every vertex represents a point in the two dimensional Euclidean plane. Associated to each edge, there is a disk placed right at its midpoint, with diameter matching the length of the edge. MASTP seeks a spanning tree of G that minimizes the area in the union of the disks associated to its edges. This paper presents Integer Programming (IP) approaches for MASTP, introduces pre-processing techniques to reduce the size of the formulations, and characterizes valid inequalities for reinforcing its Linear Programming Relaxation bounds. Several Branch-and-cut (BC) algorithms exploiting such ideas are introduced. Additionally, we also apply Benders Decomposition to one of these formulations. An accompanying BC method, that separates Benders optimality cuts, is also introduced. Aiming to save linear programming re-optimization times, that algorithm makes use of an early branching strategy. Given the set of valid inequalities used in the polyhedral representation of the problem and the best available upper bound for the optimal cost, it detects if the node cannot be pruned by bounds and then stops the cutting plane generation, in order to branch. Our algorithms manage to solve instances with up to only 15 vertices, suggesting thus that MASTP is hard to solve in practice, at least with the currently available methods. Thanks to the early branching strategy, the Benders based BC method obtained the best computational results, by far. It solved more instances to proven optimality than the other algorithms (31 out of 45 cases) and it is about three times faster than the second best performing algorithm.