To efficiently derive bounds for large-scale instances of the capacitated fixed-charge network design problem, Lagrangian relaxations appear promising. This paper presents the results of comprehensive experiments aimed at calibrating and comparing bundle and subgradient methods applied to the optimization of Lagrangian duals arising from two Lagrangian relaxations. This study substantiates the fact that bundle methods appear superior to subgradient approaches because they converge faster and are more robust relative to different relaxations, problem characteristics, and selection of the initial parameter values. It also demonstrates that effective lower bounds may be computed efficiently for large-scale instances of the capacitated fixed-charge network design problem. Indeed, in a fraction of the time required by a standard simplex approach to solve the linear programming relaxation, the methods we present attain very high quality solutions.

Abstract not found

We propose a novel approach to the study of Internet topology in which we use an optimization framework to model the mechanisms driving incremental growth. While previous methods of topology generation have focused on explicit replication of statistical properties, such as node hierarchies and node degree distributions, our approach addresses the economic tradeoffs, such as cost and performance, and the technical constraints faced by a single ISP in its network design. By investigating plausible objectives and constraints in the design of actual networks, observed network properties such as certain hierarchical structures and node degree distributions can be expected to be the natural by-product of an approximately optimal solution chosen by network designers and operators. In short, we advocate here essentially an approach to network topology design, modeling, and generation that is based on the concept of Highly Optimized Tolerance (HOT). In contrast with purely descriptive topology modeling, this opens up new areas of research that focus on the causal forces at work in network design and aim at identifying the economic and technical drivers responsible for the observed large-scale network behavior. As a result, the proposed approach should have significantly more predictive power than currently pursued efforts and should provide a scientific foundation for the investigation of other important problems, such as pricing, peering, or the dynamics of routing protocols.

In this paper a survey on existing algorithms for the capacitated minimum spanning tree problem (CMST) is given. The algorithms are classified providing some insights into their fundamental principles. Reporting the literature, comparisons of the solution quality are given. As one result of the exploration it is observed that heuristic procedures for the CMST in general consider arcs when generating or transforming a solution. Contrary to this we develop an improvement procedure which is based on partitioning nodes into subsets thus focusing more on the combinatorial nature of the CMST. Given a feasible solution the attained node assignment is altered by a local search process based on shifts and node exchanges. To overcome local optimality simulated annealing and tabu search are investigated. Computations on some benchmark test problems are reported and improvements over the well-known Esau-Williams solution are presented. Some new best solutions are obtained.

The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current state-of-the-art. In particular, continuous location models, network location models, mixed-integer programming models, and applications are summarized.

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree with an additional cardinality constraint on the sizes of the subtrees incident to a given root node. The CMST problem is an NP-complete problem, and existing exact algorithms can solve only small size problems. Currently, the best available heuristic procedures for the CMST problem are tabu search algorithms due to Amberg et al. and Sharaiha et al. These algorithms use two-exchange neighborhood structures that are based on exchanging a single node or a set of nodes between two subtrees. In this paper, we generalize their neighborhood structures to allow exchanges of nodes among multiple subtrees simultaneously; we refer to such neighborhood structures as multi-exchange neighborhood structures. Our first multiexchange neighborhood structure allows exchanges of single nodes among several subtrees. Our second multi-exchange neighborhood structure allows exchanges that involve multiple subtrees. The size of each of these neighborhood structures grows exponentially with the problem size without any substantial increase in the computational times needed to find improved neighbors. Our approach, which is based on the cyclic transfer neighborhood structure due to Thompson and Psaraftis and Thompson and Orlin transforms a profitable exchange into a negative cost subset-disjoint cycle in a graph, called an improvement graph, and identifies these cycles using variants of shortest path label-correcting algorithms. Our computational results with GRASP and tabu search algorithms based on these neighborhood structures reveal that (i) for the unit demand case our algorithms obtained the best available solutions for all benchmark instances and improved some; and (ii) for the heterogeneous deman...

The capacitated multipoint network design problem (CMNDP) is NP-complete. In this paper, a hybrid genetic algorithm for CMNDP is proposed. The multiobjective hybrid genetic algorithm (MOHGA) differs from other genetic algorithms (GA's) mainly in its selection procedure. The concept of subpopulation is used in MOHGA. Four subpopulations are generated according to the elitism reservation strategy, the shifting Prfer vector, the stochastic universal sampling, and the complete random method, respectively. Mixing these four subpopulations produces the next generation population. The MOHGA can effectively search the feasible solution space due to population diversity. The MOHGA has been applied to CMNDP. By examining computational and analytical results, we notice that the MOHGA can find most nondominated solutions and is much more effective and efficient than other multiobjective GA's.

Abstract: The advent of Sonet and DWDM mesh-restorable networks which contain explicit reserva-tions of spare capacity for restoration presents a new problem in topological network design. On the one hand, the routing of working flows wants a sparse tree-like graph for minimization of the classic fixed charge plus routing (FCR) costs. On the other hand, restorability requires a closed (bi-connected) and pref-erably high-degree topology for efficient sharing of spare capacity allocations (SCA) for restoration over non-simultaneous failure scenarios. These diametrically opposed considerations underlie the determination of an optimum physical facilities graph for a broadband network provider. Standalone instances of each constituent problem are NP-hard. The full problem of simultaneously optimizing mesh-restorable topol-ogy, routing, and sparing is therefore very difficult computationally. Following a comprehensive survey of prior work on topological design problems, we provide a {1-0} MIP formulation for the complete mesh-restorable design problem and also propose a novel three-stage heuristic. The heuristic is based on the hypothesis that the union set of edges obtained from separate FCR and SCA sub-problems constitutes an effective topology space within which to solve a restricted instance of the full problem. Where fully opti-mal reference solutions are obtainable the heuristic shows less than 8 % gaps but runs in minutes as opposed to days. In other test cases the reference problem cannot be solved to optimality and we can only

The planning of telecommunications networks raises various combinatorial optimization problems related to network design, routing of demands, multiplexing, etc. We provide in this paper an overview of such problems and a synthesis of the related literature, while introducing the field to non expert readers from both the operations research and telecommunications communities.

Whenever an edge failure occurs in a telecommunications network, the amount of traffic using the faulty edge has to be rerouted on some other paths through the network thus requiring the presence of reserve capacity in the network. This leads to the design of a network of extra capacities sufficient to independently reroute the flow on each arc and having minimum cost. This problem is known as a network synthesis problem with non-simultaneous multicommodity flow requirements. Here we propose to use a surrogate duality method based on a model formulation containing several blocks of constraints to compute lower bounds on the linear relaxation of the problem. This bound is then considered as input to a greedy heuristic to find integer feasible solutions. Finally, a tabu search procedure based on this greedy heuristic is developed and tested on a wide set of problem instances. Keywords: network synthesis, survavibility, restoration, surrogate Service de Math'emathiques de la Gestion, ...