New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller–Tucker–Zemlin constraints

Abstract

a b s t r a c tGiven an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller-Tucker-Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topologyenforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).

The hop-constrained minimum cost flow spanning tree problem with nonlinear costs: an ant colony optimization approach

“…costs associated with the usage of arcs and the costs associated with routing of the flow, as given in (1). Equation (2) guarantee that every node is in the solution in exactly one position.…”

“…We refer the reader to [21,23] for a literature review on the HMST (e.g. [1,9,[15][16][17]) and related problems. See also the works of [27] and [28] proposing ACO approaches to problems in communication networks which, although related, are different since neither arc flows nor flow requirements are considered.…”

Optimal consumed electric energy while sequencing vehicle trips in a personal rapid transit transportation system

“…These constraints are required, especially in some transportation and networking design problems, where a particular node (depot) is the starting and the ending point for any tour. In other words, any tour which doesn't contain the depot is eliminated([26], [18], [1], [3], [2], and [13] ). Consequently, any new formulation containing "Miller-Tucker-Zemlin constraints" is referred to as MTZ formulation.…”

Two dependency constrained spanning tree problems

“…This kind of lifting has been used to model the hop‐constrained MST (Gouveia, ; Akgün and Tansel, ). In such a case, however, in is reduced by a fixed value p , which represents the maximum number p of hops from the root allowed in the solution.…”