A 2‐path approach for odd‐diameter‐constrained minimum spanning and Steiner trees

Abstract

In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flow-based formulations for diameter-constrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficult to solve than those when D is even. In this article we provide an alternative modeling approach for the situation when D is odd. The approach views the diameter-constrained minimum spanning tree as being composed of a variant of a directed spanning tree (from an artificial root node) together with two constrained paths, a shortest and a longest path, from the root node to any node in the tree. We also show how to view the feasible set of the linear programming relaxation of the new formulation as the intersection of two integer polyhedra, a so-called triangle-tree polyhedron and a constrained path polyhedron. This characterization improves upon a model of Gouveia and Magnanti (2003) whose linear programming relaxation feasible set is the intersection of three rather than two integer polyhedra. The linear programming gaps for the tightened model are very small, typically less than 0.5%, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, using the new model, we have been able to solve large Euclidean problem instances that are not solvable by the previous approaches.

Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs

“…For the first group we report results given by the methods described in [33], [20], [16] (for situations with D even), [17] (for situations with D odd), and [29]. The first method uses formulations based on enhanced versions of the Miller-Tucker-Zemlin constraints.…”

“…Second, we observe that in the linear programming relaxation of the Hop-NF formulation, constraints (16), (17) when j = k and constraints (18) are satisfied as equalities (and thus, the same happens with constraints (21) and (22)). Combining these equalities gives…”

“…we obtain That is, under (44, 45) the only essential difference between the two models are constraints (17) in Hop-NF which are a disaggregated version of constraints (41) in Hop-MCF (later on, with a small example we will give some intuition on why these two sets have different modelling properties). Since adding equalities (44, 45) to Hop-NF does not modify its linear programming value, we conclude that Proposition 2 The linear programming relaxation value of the revised Hop-NF model is at least as good as the linear programming relaxation value of the Hop-MCF model.…”

Solving the Euclidean Bounded Diameter Minimum Spanning Tree Problem by Clustering–Based (Meta–)Heuristics

“…To solve this problem to proven optimality there exist various integer linear programming (ILP) approaches like hop-indexed multi-commodity network flow models [2,3] or a Branch&Cut formulation based on a more compact model but strengthened by a special class of cutting planes [4]. They all have in common that they are only applicable to relatively small instances, i.e.…”

Heuristic Algorithms for Solving Bounded Diameter Minimum Spanning Tree Problem and Its Application to Genetic Algorithm Development

“…Exact approaches for solving the BDMST problem are based on mixed linear integer programming (N.R. Achuthan et al, 1994), (L Gouveia et al, 2004). More recently, Gruber and Raidl suggested a branch and cut algorithm based on compact 0-1 integer linear programming .…”