Algorithms for a multi-level network optimization problem

In this paper we work on a multi-level network optimization problem that integrates into the same model important aspects of: (i) discrete facility location, (ii) topological network design, and (iii) network dimensioning. Potential applications for the model are discussed, stressing its growing importance. The multi-level network optimization problem treated is defined and a mathematical programming formulation is presented. We make use of a branch-and-bound algorithm based on Lagrangean relaxation lower bounds to introduce some new powerful auxiliary algorithms to exactly solve the problem. We conduct a set of computational experiments that indicate the quality of the proposed approach.     

An algorithm for network dimensioning under reliability considerations

In this paper we introduce a new methodology to adjust link capacities in circuit switched networks taking into account the costing policy and reliability considerations. This methodology, which is an extension of previous work on reliability evaluation using routing models, is based on a cyclic decomposition algorithm which alternates between a routing subproblem and a link capacity adjustment subproblem. The proposed procedure, which is shown to converge to a global optimum for the dimensioning/routing problem, has been tested on a 14 undirected arc problem for various levels of link failure probability. The numerical results are extremely satisfactory and they demonstrate the usefulness of the proposed method for proper network dimensioning.     

Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization

Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lagrangian relaxation of the POPs. This shows that existing results in this direction for quadratic POPs extend to general POPs. Also, we provide some tractable approximation strategies for CP and CPSD tensor relaxations. These approximation strategies show that, with a similar computational effort, bounds obtained from them for general POPs can be tighter than bounds for these problems obtained by reformulating the POP as a quadratic POP, which subsequently can be approximated using CP and PSD matrices. To illustrate our results, we numerically compare the bounds obtained from these relaxation approaches on small scale fourth-order degree POPs.     

Evolving System Architectures for Multimedia Network Design

This paper discusses implementation issues concerning a telecommunications planning system for networks with hypothetical multimedia stochastic traffic. Stochasticity of the traffic and probabilistic statements about service commitment are captured by a chance-constrained stochastic programming formulation of the complex network dimensioning and traffic management problem treated in our previous work. Our approach involves a hierarchy of design objectives associated with the respective network layers. These are met by constituting a set of models – the Integrated Network Design System (INDS) – with common data, solvers and a graphical user interface (GUI) operating under the interactive control of the network planner.     

A Branch-and-Bound Algorithm to Solve a Multi-level Network Optimization Problem

Multi-level network optimization problems arise in many contexts such as telecommunication, transportation, and electric power systems. A model for multi-level network design is formulated as a mixed-integer program. The approach is innovative because it integrates in the same model aspects of discrete facility location, topological network design, and dimensioning. We propose a branch-and-bound algorithm based on Lagrangian relaxation to solve the model. Computational results for randomly generated problems are presented showing the quality of our approach. We also present and discuss a real world problem of designing a two-level local access urban telecommunication network and solving it with the proposed methodology.     

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.     

Design and dimensioning of hydrogen transmission pipeline networks

This work considers the problem of the optimal design of an hydrogen transmission network. This design problem includes the topology determination and the pipelines dimensioning problem. We define a local search method that simultaneously looks for the least cost topology of the network and for the optimal diameter of each pipe. These two problems were generally solved separately these last years. The application to the case of development of future hydrogen pipeline networks in France has been conducted at the local, regional and national levels. We compare the proposed approach with another using Tabu search heuristic.     

A permutation-coded evolutionary strategy for multi-objective GSM network planning

The base station placement problem, with n potential candidate sites is NP-Hard with 2 ⁿ solutions (Mathar and Niessen, Wirel. Netw. 6, 421–428, 2000). When dimensioned on m unknown variable settings (e.g., number of power settings + number of tilt settings, etc.) the computational complexity becomes (m+1) ⁿ (Raisanen, PhD. thesis, 2006). We introduce a novel approach to reduce the computational complexity by dimensioning sites only once to guarantee traffic hold requirements are satisfied. This approach works by determining the maximum set of service test points candidate sites can handle without exceeding a hard traffic constraint, T _MAX . Following this, the ability of two evolutionary strategies (binary and permutation-coded) to search for the minimum set cover are compared. This reverses the commonly followed approach of achieving service coverage first and then dimensioning to meet traffic hold. To test this approach, three realistic GSM network simulation environments are engineered, and a series of tests performed. Results indicate this approach can quickly meet network operator objectives.     

Metaheuristic hybridizations for the regenerator placement and dimensioning problem in sub-wavelength switching optical networks

Physical layer impairments severely limit the reach and capacity of optical systems, thereby hampering the deployment of transparent optical networks (i.e., no electrical signal regenerators are required). Besides, the high cost and power-consumption of regeneration devices makes it unaffordable for network operators to consider the opaque architecture (i.e., regeneration is available at every network node). In this context, translucent architectures (i.e., regeneration is only available at selected nodes) have emerged as the most promising short term solution to decrease costs and energy consumption in optical backbone networks. Concurrently, the coarse granularity and inflexibility of legacy optical technologies have re-fostered great interest in sub-wavelength switching optical networks, which introduce optical switching in the time domain so as to further improve resources utilization. In these networks, the complex regenerator placement and dimensioning problem emerges. In short, this problem aims at minimizing the number of electrical regenerators deployed in the network. To tackle it, in this paper both a greedy randomized adaptive search procedure and a biased random-key genetic algorithm are developed. Further, we enhance their performance by introducing both path-relinking and variable neighborhood descent as effective intensification procedures. The resulting hybridizations are compared among each other as well as against results from optimal and heuristic mixed integer linear programming formulations. Illustrative results over a broad range of network scenarios show that the biased random-key genetic algorithm working in conjunction with these two intensification mechanisms represents a compelling network planning algorithm for the design of future sub-wavelength optical networks.     

A polynomial optimization approach to constant rebalanced portfolio selection

We address the multi-period portfolio optimization problem with the constant rebalancing strategy. This problem is formulated as a polynomial optimization problem (POP) by using a mean-variance criterion. In order to solve the POPs of high degree, we develop a cutting-plane algorithm based on semidefinite programming. Our algorithm can solve problems that can not be handled by any of known polynomial optimization solvers.     

Combining very large scale and ILP based neighborhoods for a two-level location problem

In this paper we tackle a generalization of the Single Source Capacitated Facility Location Problem in which two sets of facilities, called intermediate level and upper level facilities, have to be located; the dimensioning of the intermediate set, the assignment of clients to intermediate level facilities, and of intermediate level facilities to upper level facilities, must be optimized, as well. Such problem arises, for instance, in telecommunication network design: in fact, in hierarchical networks the traffic arising at client nodes often have to be routed through different kinds of facility nodes, which provide different services. We propose a heuristic approach, based on very large scale neighborhood search to tackle the problem, in which both ad hoc algorithms and general purpose solvers are applied to explore the search space. We report on experimental results using datasets from the capacitated location literature. Such results show that the approach is promising and that Integer Linear Programming based neighborhoods are significantly effective.     

The cut condition for robust network design

We consider the Robust Network Design problem, i.e. the problem of dimensioning the capacities on the edges of a graph to serve point-to-point traffic demands subject to uncertainty. The scope of the paper is to investigate when the cut condition is sufficient to guarantee the feasibility for such problem.     

Single-path routing of stochastic flows in networks

The paper revisits a very simple network routing and dimensioning model, with two specific assumptions: the traffic amounts to be routed are Gaussian random variables, and each commodity must use one single route in the network. The need to control congestion leads naturally to probabilistic constraints. The impact of stochastic assumptions on solution algorithms is investigated, when compared to the usual deterministic case.     

The convex hull of two core capacitated network design problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost.This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.Research of this author was supported in part by a Faculty Grant from the Katz Graduate School of Business, University of Pittsburgh.     

Virtual path layouts optimizing total hop count on ATM tree networks

This paper presents some algorithmic results concerning virtual path layouts for the one-to-many communication problem in ATM tree networks. The ATM network model is based on covering the network with a layout of virtual paths, under some constraints on the allowed load, namely, the number of paths that can share an edge. The quality measure used is the hop count, namely, the number of edges traversed between two vertices that need to communicate. Whereas most former results concerned the maximum hop count of the virtual path layout, our interest here is in measuring its total hop count, or alternatively its average hop count. The paper presents a dynamic programming algorithm for planning ATM network layouts with minimal total hop count for one-to-many requirements under load constraints over the class of tree networks.     

Turing Machines Connected to the Undecidability of the Halting Problem

The problem of finding a Turing machine with undecidable halting problem whose program contains the smallest number of instructions is well known. Obviously, such a machine must satisfy the following condition: by deleting even a single instruction from its program, we get a machine with decidable halting problem. In this paper, Turing machines with undecidable halting problem satisfying this condition are called connected. We obtain a number of general properties of such machines and deduce their simplest corollaries concerning the minimal machine with undecidable halting problem.     

A mathematical program to refine gene regulatory networks

In this paper, we propose a methodology for making sense of large, multiple time-series data sets arising in expression analysis. Specifically, we present a mathematical model to release a reduced and coherent regulatory system given a putative regulatory network. We give two equivalent formulations of the problem and prove that the problem is NP-complete. For solving large scale instances we implemented an Ant Colony Optimization procedure. A computational analysis on randomly generated test instances validates the proposed algorithm and the computations on real data concerning Saccharomyces cerevisiae show the practicability of the proposed methodology.     

Commuting graphs of boundedly generated semigroups

Araújo, Kinyon and Konieczny (2011) pose several problems concerning the construction of arbitrary commuting graphs of semigroups.We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems.We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two.We also address a problem concerning knit degree.     

On the theorem of Barban and Davenport-Halberstam in algebraic number fields

The Barban-Davenport-Halberstam theorem is generalized to arbitrary algebraic number fields of finite degree over the rationals. As an application, we consider a divisor problem and a result of Elliott and Halberstam concerning the least prime in an arithmetic progression.     

Interaction between data analysis and telecommunications

We consider interaction between data analysis and telecommunications from both sides, examining the contribution of each discipline to the other. The fast changes in telecommunications have some important consequences. First, the importance of statistics in the design of a network is growing, given the evolution of technology towards more digitalization and the resulting changes in dimensioning the network. Second, the technical tools of statistics are deeply modified, and there is concern about how to take this into account—particularly how to take advantage of the new economy of ‘value added services’ which uses the network as a ‘market place’.