The Steiner connectivity problem

Mathematical Programming, 2011

The hop-constrained minimum spanning tree problem (HMSTP) is an NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner tree problem (STP) in an appropriate layered graph. We prove that the directed cut model for the STP defined in the layered graph, dominates the best previously known models for the HMSTP. We also show that the Steiner directed cuts in the extended layered graph space can be viewed as being a stronger version of some previously known HMSTP cuts in the original design space. Moreover, we show that these strengthened cuts can be combined and projected into new families of cuts, including facet defining ones, in the original design space. We also adapt the proposed approach to the diameter-constrained minimum spanning tree problem (DMSTP). Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.

is work deals with the creation and optimization of real-world instances in the design of telecommunication networks. We propose two new sets of benchmark instances for the Steiner tree problem in graphs, which is one of the fundamental network optimization problems. Our instances are large, sparse graphs that contain over 100 000 nodes and originate from real-world applications of deploying last-mile fiber-optic networks. To obtain a rough estimate on the hardness of the new instances, we measured the performance of preprocessing techniques and of an exact algorithm based on branch-and-cut. is work shall establish a missing link between the real world and the mathematical modeling and optimization of telecommunication networks.

HAL (Le Centre pour la Communication Scientifique Directe), 2013

We address the Steiner tree problem with revenues, budget and hop constraints (STPRBH), which is a generalization of the well-known Steiner tree problem. Given a connected undirected graph, a root node, edge costs and delays, nodes revenues, as well as a preset budget and hop, the STPRBH seeks to …nd a subtree that includes the root node, satis…es bound constraints on the total edge cost as well as the number of edges between any node and the root node, while maximizing the sum of the total node revenues. We focus on investigating polynomial-sized formulations. First, we propose an enhanced formulation based on the Miller-Tucker-Zemlin subtour constraints. Next, we investigate a nonlinear MIP formulation that is linearized using the Reformulation-Linearization Technique (RLT). We present the results of a comprehensive computational study of the proposed formulations. These result provide evidence that benchmark instances with up to 500 nodes can be e¤ectively solved using the proposed RLT-based formulation.

2009

Abstract The Steiner tree problem with revenues, budget and hop constraints is a variant of the Steiner tree problem with two main modifications:(a) besides the costs associated with arcs, there are also revenues associated with the vertices, and (b) there are additional budget and hop constraints, which impose limits on the total cost of the network and on the number of edges between any vertex and the root, respectively.

Annals of Operations Research, 2015

A complete weighted graph, G(X, Γ, W), is considered. LetX ⊂ X be some subset of vertices and, by definition, a Steiner tree is any tree in the graph G such that the set of the tree vertices includes setX. The Steiner tree problem consists of constructing the minimum-length Steiner tree in graph G, for a given subset of verticesX The effectively computable estimate of the minimal Steiner tree is obtained in terms of the mean value and the variance over the set of all Steiner trees. It is shown that in the space of the lengths of the graph edges, there exist some regions where the obtained estimate is better than the minimal spanning tree-based one.

European Journal of Operational Research, 2010

This paper considers a tricriteria Steiner Tree Problem arising in the design of telecommunication networks. The objective functions consist of maximizing the revenue and of minimizing the maximal distance between each pair of interconnected nodes, as well as the maximal number of arcs between the root and each node. A polynomial algorithm is developed for the generation of a minimal complete set of Pareto-optimal Steiner trees. Optimality proofs are given and computational experience on a set of randomly generated problems is reported.

We design combinatorial approximation algorithms for the Capacitated Steiner Network (Cap-SN) problem and the Capacitated Multicommodity Flow (Cap-MCF) problem. These two problems entail satisfying connectivity requirements when edges have costs and hard capacities. In Cap-SN, the flow has to be supported separately for each commodity while in Cap-MCF, the flow has to be sent simultaneously for all commodities. We show that the Group Steiner problem on trees ([12]) is a special case of both problems. This implies the first polylogarithmic lower bound for these problems by [17]. We then give various approximations to special cases of the problems. We generalize the well known Source location problem (see for example [19]), to a natural problem called the Connected Rent or Buy Source Location problem. We show that this problem is a a simplification of Cap-SN and Cap-MCF and a generalization of Group Steiner on general graphs. We use Group Steiner Tree techniques, and more sophisticated techniques, to derive log 3+ n approximation for the Connected Rent or Buy Source Location problem which is close to the best approximation known for Group Steiner on general graphs. Another special case we study is as follows. Given a bipartite graph G = (A ∪ B, E) and an integer k > 0, find A ⊆ A and B ⊆ B of minimum total size |A | + |B | such that there exist k edge-disjoint paths in G from vertices in A to vertices in B. This problem is a special case of the Steiner Network problem with vertex costs [20]. In [20] Nutov asked the open question if the Steiner network problem with vertex costs admits an o(k) ratio. We give an o(k) approximation for this special case, which could be a step toward resolving the open question of Nutov. We provide an O(√ k log k) approximation ratio for the problem. We also show that we can compute a solution of optimum value, while being able to route O(k/polylog n) flow, where n is Part of this work was done at DIMACS. We thank DIMACS for their hospitality.

We present a specialized branch-and-bound (b&b) algorithm for the Euclidean Steiner tree problem (ESTP) in R n and apply it to a convex mixed-integer nonlinear programming (MINLP) formulation of the problem, presented by Fampa and Maculan. The algorithm contains procedures to avoid difficulties observed when applying a b&b algorithm for general MINLP problems to solve the ESTP. Our main emphasis is on isomorphism pruning, in order to prevent solving several equivalent subproblems corresponding to isomorphic Steiner trees. We introduce the concept of representative Steiner trees, which allows the pruning of these subproblems, as well as the implementation of procedures to fix variables and add valid inequalities. We also propose more general procedures to improve the efficiency of the b&b algorithm, which may be extended to the solution of other MINLP problems. Computational results demonstrate substantial gains compared to the standard b&b for convex MINLP.

Given an undirected weighted graph G = (V, E, c) and a set T , where V is the set of nodes, E is the set of edges, c is a cost function, and T is a subset of nodes called terminals, the Steiner tree problem in graphs is that of finding the subgraph of the minimum weight that connects all of terminals. The Steiner tree problem is an example of an NP-complete combinatorial optimization problem [1]. Thus, approximate methods are usually employed for constructing the Steiner tree. In this study, the KMB algorithm [2], which is an efficient construction method for Steiner tree problems, is enhanced by considering edge betweenness [3]. The results of numerical simulations indicate that our improved KMB algorithm shows good performances for various types of benchmark Steiner tree problems.