Geir Dahl

A Cutting Plane Algorithm for Multicommodity Survivable Network Design Problems

We present a cutting plane algorithm for solving the following telecommunications network design problem: given point-to-point traffic demands in a network, specified survivability requirements and a discrete cost/capacity function for each link, find minimum cost capacity expansions satisfying the given demands. This algorithm is based on the polyhedral study described in [19]. In this article we describe the underlying problem, the model and the main ingredients in our algorithm. This includes: initial formulation, feasibility test, separation for strong cutting planes, and primal heuristics. Computational results for a set of real-world problems are reported.

Properties of the beampattern of weight- and layout-optimized sparse arrays

Theory for random arrays predicts a mean sidelobe level given by the inverse of the number of elements. In practice, however, the sidelobe level fluctuates much around this mean. In this paper two optimization methods for thinned arrays are given: one is for optimizing the weights of each element, and the other one optimizes both the layout and the weights. The weight optimization algorithm is based on linear programming and minimizes the peak sidelobe level for a given beamwidth. It is used to investigate the conditions for finding thinned arrays with peak sidelobe level at or below the inverse of the number of elements. With optimization of the weights of a randomly thinned array, it is possible to come quite close and even below this value, especially for 1D arrays. Even for 2D sparse arrays a large reduction in peak sidelobe level is achieved. Even better solutions are found when the thinning pattern is optimized also. This requires an algorithm that uses mixed integer linear programming. In this case solutions, with lower peak sidelobe level than the inverse number of elements can be found both in the 1D and the 2D cases.

On Formulations and Methods for the Hop-Constrained Minimum Spanning Tree Problem

In this chapter we present a general framework for modeling the hopconstrained minimum spanning tree problem (HMST) which includes formulations already presented in the literature. We present and survey different ways of computing a lower bound on the optimal value. These include, Lagrangian relaxation, column generation and model reformulation. We also give computational results involving instances with 40 and 80 nodes in order to compare some of the ideas discussed in the chapter.

The 2-hop spanning tree problem

A spanning tree in a graph G where each node has distance at most 2 from a root node r is called a 2-hop spanning tree. For given edge weights the 2-hop spanning tree problem is to find a minimum weight 2-hop spanning tree. The problem is NP-hard. We study the problem from a polyhedral point of view based on a directed formulation and give, for example, a completeness result when G is an n-wheel.

Optimization and reconstruction of hv-convex (0,1)-matrices

We consider a variant of the NP-hard problem of reconstructing hv-convex (0, 1)-matrices from known row and column sums. Instead of requiring the ones to occur consecutively in each row and column, we maximize the number of neighboring ones. This is reformulated as an integer programming problem. A solution method based on variable splitting is proposed and tested with good results on moderately sized test problems.

Notes on polyhedra associated with hop-constrained paths

We study the dominant of the convex hull of st-paths with at most k edges in a graph. A complete linear description is obtained for k= =4 is given.

Routing Through Virtual Paths in Layered Telecommunication Networks

We study a network configuration problem in telecommunications where one wants to set up paths in a capacitated network to accommodate given point-to-point traffic demand. The problem is formulated as an integer linear programming model where 0-1 variables represent different paths. An associated integral polytope is studied, and different classes of facets are described. These results are used in a cutting plane algorithm. Computational results for some realistic problems are reported.

The cardinality-constrained shortest path problem in 2-graphs

We study the cardinality-constrained shortest path problem in acyclic graphs and, in particular, in the class of 2-graphs where we show that the problem may be solved by linear programming. A combinatorial algorithm is introduced based on some adjacency results for associated polytopes. An application in curve approximation is discussed and computational results are given where the mentioned algorithms are compared to Lagrangian relaxation and dynamic programming algorithms.

Lagrangian-based methods for finding MAP solutions for MRF models

Finding maximum a posteriori (MAP) solutions from noisy images based on a prior Markov random field (MRF) model is a huge computational task. In this paper, we transform the computational problem into an integer linear programming (ILP) problem. We explore the use of Lagrange relaxation (LR) methods for solving the MAP problem. In particular, three different algorithms based on LR are presented. All the methods are competitive alternatives to the commonly used simulation-based algorithms based on Markov Chain Monte Carlo techniques. In all the examples (including both simulated and real images) that have been tested, the best method essentially finds a MAP solution in a small number of iterations. In addition, LR methods provide lower and upper bounds for the posterior, which makes it possible to evaluate the quality of solutions and to construct a stopping criterion for the algorithm. Although additive Gaussian noise models have been applied, any additive noise model fits into the framework.

On the k edge-disjoint 2-hop-constrained paths polytope

The k edge-disjoint 2-hop-constrained paths problem consists in finding a minimum cost subgraph such that between two given nodes s and t there exist at least k edge-disjoint paths of at most 2 edges. We give an integer programming formulation for this problem and characterize the associated polytope.