Network Flow Models for Designing Diameter-Constrained Minimum Spanning and Steiner Trees

The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We formulate the problem as a directed tree from a selected central node or a selected central edge. Our model simultaneously finds a central node or a central edge and uses it as the source for the commodities in a directed multicommodity flow model with hop constraints. The new model has been able to solve the 20-node, 100-edge instance to optimality after less than four seconds. We also present model enhancements when the diameter bound is odd (these situations are more difficult). We show that the linear programming relaxation of the best formulations discussed in this paper always give an optimal integer solution for two special, polynomially-solvable cases of the problem. We also examine the Diameter Constrained Minimum Steiner Tree problem. We present computational experience in solving problem instances with up to 100 nodes and 1000 edges. The largest model contains more than 250,000 integer variables and more than 125,000 constraints

Computational results for Constrained Minimum Spanning Trees in Flow Networks

In this work, we address the problem of finding a minimum cost spanning tree on a single source flow network. The tree must span all vertices in the given network and satisfy customer demands at a minimum cost. The total cost is given by the summation of the arc setup costs and of the nonlinear flow routing costs over all used arcs. Furthermore, we restrict the trees of interest by imposing a maximum number of arcs on the longest arc emanating from the single source vertex. We propose a dynamic programming model an solution procedure to solve this problem exactly. Intensive computational experiments were performed using randomly generated test problems and the results obtained are reported. From them we can conclude that the method performance is independent of the type of cost functions considered and improves with the tightness of the constrains.Dynamic programming, network flows, constrained trees, general nonlinear costs

Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

Target-based Distributionally Robust Minimum Spanning Tree Problem

Due to its broad applications in practice, the minimum spanning tree problem and its all kinds of variations have been studied extensively during the last decades, for which a host of efficient exact and heuristic algorithms have been proposed. Meanwhile, motivated by realistic applications, the minimum spanning tree problem in stochastic network has attracted considerable attention of researchers, with respect to which stochastic and robust spanning tree models and related algorithms have been continuingly developed. However, all of them would be either too restricted by the types of the edge weight random variables or computationally intractable, especially in large-scale networks. In this paper, we introduce a target-based distributionally robust optimization framework to solve the minimum spanning tree problem in stochastic graphs where the probability distribution function of the edge weight is unknown but some statistical information could be utilized to prevent the optimal solution from being too conservative. We propose two exact algorithms to solve it, based on Benders decomposition framework and a modified classical greedy algorithm of MST problem (Prim algorithm),respectively. Compared with the NP-hard stochastic and robust spanning tree problems,The proposed target-based distributionally robust minimum spanning tree problem enjoys more satisfactory algorithmic aspect and robustness, when faced with uncertainty in input data