Decision support system for planning telecommunication networks: a case study applied to the Andalusian region

Network planning is essential to design real Broadband Integrated Services Digital Networks (B-ISDN). This paper presents an OR application to the design of an optic fibre network for the andalusian region. The economical appraisal is the main consideration in order to take the appropriate decisions: hub location, region sizes and selection of the urban nodes that will receive telecommunication contents. A Decision Support System with a graphic interface that allows interactive analysis of different scenarios is presented. The system contains a set of mathematical programming models and it has the capability to dynamically construct and solve instances of those models. In addition, it provides data preparation and reports. The system is an integrated, user-friendly and powerful tool to make planning studies by firms developing cable network systems in the telecommunications market

Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions

Dynamic programming on tree decompositions is a frequently used approach to solve otherwise intractable problems on instances of small treewidth. In recent work by Bodlaender et al., it was shown that for many connectivity problems, there exist algorithms that use time, linear in the number of vertices, and single exponential in the width of the tree decomposition that is used. The central idea is that it suffices to compute representative sets, and these can be computed efficiently with help of Gaussian elimination. In this paper, we give an experimental evaluation of this technique for the Steiner Tree problem. A comparison of the classic dynamic programming algorithm and the improved dynamic programming algorithm that employs the table reduction shows that the new approach gives significant improvements on the running time of the algorithm and the size of the tables computed by the dynamic programming algorithm, and thus that the rank based approach from Bodlaender et al. does not only give significant theoretical improvements but also is a viable approach in a practical setting, and showcases the potential of exploiting the idea of representative sets for speeding up dynamic programming algorithms

A Comparison of Steiner Tree Relaxations

There are many (mixed) integer programming formulations of the Steiner problem in networks. The corresponding linear programming relaxations are of great interest particularly, but not exclusively, for computing lower bounds; but not much has been known ab out the relative quality of these relaxations. We compare all classical and some new relaxations from a theoretical point of view with respect to their optimal values. Among other things, we prove that the optimal value of a flowclass relaxation (e.g. the multicommodity flow or the dicut relaxation) cannot be worse than the optimal value of a tree-class relaxation (e.g. degree-constrained spanning tree relaxation) and that the ratio of the corresponding optimal values can be arbitrarily large. Furthermore, we present a new flow based relaxation, which is to the authors' knowledge the strongest linear relaxation of polynomial size for the Steiner problem in networks

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

Survivable Networks, Linear Programming Relaxations and the Parsimonious Property

We consider the survivable network design problem - the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem

On some cost allocation problems in communication networks

New technologies prompted an explosion in the development of communication networks. Modern network optimization techniques usually lead to a design of the most profitable, or the least cost network that will provide some service to customers. There are various costs and gains associated with building and using a communication network. Moreover, the involved multiple network users and/or owners possibly have conflicting objectives. However, they might cooperate in order to decrease their joint cost or increase their joint profit. Clearly, these individuals or organizations will support a globally \u27attractive\u27 solution(s) only if their expectations for a \u27fair share\u27 of the cost or profit are met. Consequently, providing network developers, users and owners with efficiently computable \u27fair\u27 cost allocation solution procedures is of great importance for strategic management. This work is an overview of some recent results (some already published as well as some new) in the development of cooperative game theory based mechanisms to efficiently compute \u27attractive\u27 cost allocation solutions for several important classes of communication networks

Computing Near-Optimal Solutions to the Steiner Problem in a Graph Using a Genetic Algorithm

A new Genetic Algorithm (GA) for the Steiner Problem in a Graph (SPG) is presented. The algorithm is based on a bitstring encoding. A bitstring specifies selected Steiner vertices and the corresponding Steiner tree is computed using the Distance Network Heuristic. This scheme ensures that every bitstring correspond to a valid Steiner tree and thus eliminates the need for penalty terms in the cost function. The GA is tested on all SPG instances from the OR-Library of which the largest graphs have 2,500 vertices and 62,500 edges. When executed 10 times on each of 58 graph examples, the GA finds the global optimum at least once for 55 graphs and every time for 43 graphs. In total the GA finds the global optimum in 77 % of all program executions and is within 1 % from the global optimum in more than 92 % of all executions. The performance is compared to that of two branch-and-cut algorithms and one of the very best deterministic heuristics, an iterated version of the Shortest Path Heuristic (SPH-I). For all test examples but one, even the worst result ever found by the GA is equal to or better than the result of SPH-I and in many cases the average error ratio of the GA is an order of magnitude better than that of SPH-I. The runtime of the GA is moderate for all test examples. This is in contrast to SPH-I as well as the branch-and-cut algorithms, for which the runtime in some cases are extremely high

Distance Transformation for Network Design Problems

International audienceWe propose a new generic way to construct extended formulations for a large class of network design problems with given connectivity requirements. The approach is based on a graph transformation that maps any graph into a layered graph according to a given distance function. The original connectivity requirements are in turn transformed into equivalent connectivity requirements in the layered graph. The mapping is extended to the graphs induced by fractional vectors through an extended linear integer programming formulation. While graphs induced by binary vectors are mapped to isomorphic layered graphs, those induced by fractional vectors are mapped to a set of graphs having worse connectivity properties. Hence, the connectivity requirements in the layered graph may cut off fractional vectors that were feasible for the problem formulated in the original graph. Experiments over instances of the Steiner Forest and Hop-constrained Survivable Network Design problems show that significant gap reductions over the state-of-the art formulations can be obtained