Minimum Spanning trees with Neighborhoods

TL;DR

This paper addresses the challenge of constructing minimum spanning trees when vertex locations are uncertain but known to lie within specific neighborhoods, proposing new mathematical models and solution methods for this problem.

Contribution

It introduces two mixed integer nonlinear programming formulations and a Benders-like decomposition approach for MSTs with neighborhood uncertainty.

Findings

Effective formulations for MSTs with neighborhoods.

A novel solution scheme with proven computational efficiency.

Ability to solve larger instances with a mathheuristic.

Abstract

This paper studies Minimum Spanning Trees under incomplete information for its vertices. We assume that no information is available on the precise placement of vertices so that it is only known that vertices belong to some neighborhoods that are second order cone representable and distances are measured with a $\ell_q$-norm. Two mixed integer non linear mathematical programming formulations are presented, based on alternative representations of subtour elimination constraints. A solution scheme is also proposed, resulting from a reformulation suitable for a Benders-like decomposition, which is embedded within an exact branch-and-cut framework. Furthermore, a mathheuristic is developed, which alternates in solving convex subproblems in different solution spaces, and is able to solve larger instances. The results of extensive computational experiments are reported and analyzed.