90C10 Integer programming
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Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or $\frac{1}{2}$, such cuts are known as $\{0,\frac{1}{2}\}$-cuts. It has been proven by Caprara and Fischetti (1996) that separation of $\{0,\frac{1}{2}\}$-cuts is NP-hard. In this paper, we study ways to separate $\{0,\frac{1}{2}\}$-cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated $\{0,\frac{1}{2}\}$-cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.
In this paper, we study the minimum converter wavelength assignment problem in optical networks. To benchmark the quality of solutions obtained by heuristics, we derive an integer programming formula tion by generalizing the formulation of Mehrotra and Trick (1996) for the vertex coloring problem. To handle the exponential number of variables, we propose a column generation approach. Computational experiments show that the value of the linear relaxation states a good lower bound and can often prove optimality of the best solution generated heuristically.
Linear Programming Lower Bounds for Minimum Converter Wavelength Assignment in Optical Networks
(2004)
In this paper, we study the conflict-free assignment of wavelengths to lightpaths in an optical network with the opportunity to place wavelength converters. To benchmark heuristics for the problem, we develop integer programming formulations and study their properties. Moreover, we study the computational performance of the column generation algorithm for solving the linear relaxation of the most promising formulation. In many cases, a non-zero lower bound on the number of required converters is generated this way. For several instances, we in fact prove optimality since the lower bound equals the best known solution value.
Stable multi-sets are an evident generalization of the well-known stable sets. As integer programs, they constitute a general structure which allows for a wide applicability of the results. Moreover, the study of stable multi-sets provides new insights to well-known properties of stable sets. In this paper, we continue our investigations started in [{\sl Koster and Zymolka 2002}] and present results of three types: on the relation to other combinatorial problems, on the polyhedral structure of the stable multi-set polytope, and on the computational impact of the polyhedral results. First of all, we embed stable multi-sets in a framework of generalized set packing problems and point out several relations. The second part discusses properties of the stable multi-set polytope. We show that the vertices of the linear relaxation are half integer and have a special structure. Moreover, we strengthen the conditions for cycle inequalities to be facet defining, show that the separation problem for these inequalities is polynomial time solvable, and discuss the impact of chords in cycles. The last result allows to interpret cliques as cycles with many chords. The paper is completed with a computational study to the practical importance of the cycle inequalities. The computations show that the performance of state-of-the-art integer programming solvers can be improved significantly by including these inequalities.
Signaling is crucial to the operation of modern telecommunication networks. A breakdown in the signaling infrastructure typically causes customer service failures, incurs revenue losses, and hampers the company image. Therefore, the signaling network has to be highest reliability and survivability. This in particular holds for the routers in such a network, called \textit{signaling transfer points\/} (STPs). The robustness of an STP can be improved by equally distributing the load over the internal processing units. Several constraints have to be taken into account. The load of the links connected to a processing unit changes over time introducing an imbalance of the load. In this paper, we show how integer linear programming can be applied to reduce the imbalance within an STP, while keeping the number of changes small. Two alternative models are presented. Computational experiments validate the integer programming approach in practice. The GSM network operator E-Plus saves substantial amounts of time and money by employing the proposed approach.
We consider the design of transparent optical networks from a practical perspective. Network operators aim at satisfying the communication demands at minimum cost. Such an optimization involves three interdependent planning issues: the dimensioning of the physical topology, the routing of lightpaths, and the wavelength assignment. Further topics include the reliability of the configuration and sparse wavelength conversion for efficient use of the capacities. In this paper, we investigate this extensive optical network design task. Using a flexible device-based model, we present an integer programming formulation that supports greenfield planning as well as expansion planning on top of an existing network. As solution method, we propose a suitable decomposition approach that separates the wavelength assignment from the dimensioning and routing. Our method in particular provides a lower bound on the total cost which allows to rate the solution quality. Computational experiments on realistic networks approve the solution approach to be appropriate.
Stable Multi-Sets
(2000)
In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets. For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined. The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks.
In this paper we describe the results of a computational study towards the (re)optimization of signaling transfer points (STPs) in telecommunication networks. The best performance of an STP is achieved whenever the traffic load is evenly distributed among the internal components. Due to the continuously changing traffic pattern, the load of the components has to be re-optimized on a regular basis. Besides the balancing objective also the number of rearrangements have to be taken into account. In this paper we present two alternative formulations to deal with both requirements. Computational results show that for both formulations (near) optimal solutions can be obtained within reasonable time limits.