Thomas M. Liebling

50 Years of Integer Programming 1958-2008

I The Early Years.- Solution of a Large-Scale Traveling-Salesman Problem.- The Hungarian Method for the Assignment Problem.- Integral Boundary Points of Convex Polyhedra.- Outline of an Algorithm for Integer Solutions to Linear Programs An Algorithm for the Mixed Integer Problem.- An Automatic Method for Solving Discrete Programming Problems.- Integer Programming: Methods, Uses, Computation.- Matroid Partition.- Reducibility Among Combinatorial Problems.- Lagrangian Relaxation for Integer Programming.- Disjunctive Programming.- II From the Beginnings to the State-of-the-Art.- Polyhedral Approaches to Mixed Integer Linear Programming.- Fifty-Plus Years of Combinatorial Integer Programming.- Reformulation and Decomposition of Integer Programs.- III Current Topics.- Integer Programming and Algorithmic Geometry of Numbers.- Nonlinear Integer Programming.- Mixed Integer Programming Computation.- Symmetry in Integer Linear Programming.- Semidefinite Relaxations for Integer Programming.- The Group-Theoretic Approach in Mixed Integer Programming.

An evolutionary heuristic for quadratic 0-1 programming

Abstract In this paper we present a heuristic algorithm for the well-known Unconstrained Quadratic 0–1 Programming Problem. The approach is based on combining solutions in a genetic paradigm and incorporates intensification algorithms used to improve solutions and speed up the method. Extensive computational experiments on instances with up to 500 variables are presented and we compare our approach both with powerful heuristic and exact algorithms from the literature establishing the effectiveness of the method in terms of solutions quality and computing time.

Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron

In this paper, we investigate the applicability of backtrack technique to solve the vertex enumeration problem and the face enumeration problem for a convex polyhedron given by a system of linear inequalities. We show that there is a linear-time backtrack algorithm for the face enumeration problem whose space complexity is polynomial in the input size, but the vertex enumeration problem requires a backtrack algorithm to solve a decision problem, called the restricted vertex problem, for each output, which is shown to be NP-complete. Some related NP-complete problems associated with a system of linear inequalities are also discussed, including the optimal vertex problems for polyhedra and arrangements of hyperplanes.

Decision-Aiding Methodology for the School Bus Routing and Scheduling Problem

We consider the school bus routing and scheduling problem, where transportation demand is known and bus scheduling can be planned in advance. We present a comprehensive methodology designed to support the decision of practitioners. We first propose a modeling framework where the focus is on optimizing the level of service for a given number of buses, then we describe an automatic procedure generating a solution to the problem. The procedure first builds a feasible solution, which is subsequently improved using a heuristic. We analyze two important issues associated with this methodology. On the one hand, we analyze the performance of three types of heuristics both on real and synthetic data. We recommend the use of a simulated annealing technique exploring infeasible solutions, which performs slightly better than all others. More importantly, we find that the performance of all heuristics is not globally affected by the choice of the parameters. This is important from a practitioner viewpoint, because the fine-tuning of algorithm parameters is not critical for the algorithms performance. We have successfully applied our methods on real problems and on large-scale problems. On the other hand, we propose an interactive tool allowing the practitioner to visualize the proposed solution, to test its robustness, and to dynamically rebuild new solutions if the data of the original problem are modified.

A polynomial case of unconstrained zero-one quadratic optimization

Abstract.Unconstrained zero-one quadratic maximization problems can be solved in polynomial time when the symmetric matrix describing the objective function is positive semidefinite of fixed rank with known spectral decomposition.

Tracking elementary particles near their primary vertex: a combinatorial approach

Colliding beams experiments in High Energy Physics rely on solid state detectors to track the flight paths of charged elementary particles near their primary point of interaction. Reconstructing tracks in this region requires, per collision, a partitioning of up to 103 highly correlated observations into an unknown number of tracks. We report on the successful implementation of a combinatorial track finding algorithm to solve this pattern recognition problem in the context of the ALEPH experiment at CERN. Central to the implementation is a 5-dimensional axial assignment model (AP5) encompassing noise and inefficiencies of the detector, whose weights of assignments are obtained by means of an extended Kalman filter. A preprocessing step, involving the clustering and geometric partitioning of the observations, ensures reasonable bounds on the size of the problems, which are solved using a branch & bound algorithm with LP relaxation. Convergence is reached within one second of CPU time on a RISC workstation in average.

The Laguerre model of grain growth in two dimensions I. Cellular structures viewed as dynamical Laguerre tessellations

The basic concepts of a deterministic model for simulating grain growth or foam coarsening in two dimensions are presented, and the fundamental tools necessary for its implementation into computer codes are provided. It is assumed that the actual cell structure can be satisfactorily represented by a Laguerre (or weighted Voronoi) diagram, entirely determined by a collection of weighted sites or circles. The cellular motion (including the induced elementary topological transformations) is subjected to the motion of this collection, allowing for the design of robust and efficient algorithms. An example of a simple motion equation, previously used in vertex models of grain growth, is reformulated to be operational for the collection of the circles generating Laguerre diagrams. The model as a whole can be extended in three dimensions without principle difficulty. Part II of the paper will report and discuss simulation results obtained in two dimensions.

The Laguerre model for grain growth in three dimensions

We propose a novel and poweful methodology for three- dimensional (3D) grain growth modelling in both the anisotropic and the locally inhomogeneous cases, thus significantly generalizing previous related two-dimensional work. The fundamental modelling structures for the polycrystals and their behaviour are dynamically evolving Laguerre diagrams in the flat 3D torus. The weighted generating sites of such diagrams obey motion equations ensuing from system interface energy minimization with consequent evolution of the associated structures and resulting in elementary topological transformations thereof. We have implemented the models in state-of-the-art computer simulation codes and made large-scale runs assuming either constant (isotropic) or variable (anisotropic) interface specific energies. In both cases, the simulated evolution reproduced the main features of the normal grain growth process in polycrystalline materials, that is the grain growth power law, typical distributions of grain sizes and shapes, and the scaling behaviour in long-term regime. The simulated distribution of grain-to- grain misorientation and its evolution in a simple hypothetical case have also been obtained for the first time.

Voronoi diagrams on piecewise flat surfaces and an application to biological growth

This paper introduces the notion of Voronoi diagrams and Delaunay triangulations generated by the vertices of a piecewise flat, triangulated surface. Based on properties of such structures, a generalized flip algorithm to construct the Delaunay triangulation and Voronoi diagram is presented. An application to biological membrane growth modeling is then given. A Voronoi partition of the membrane into cells is maintained during the growth process, which is driven by the creation of new cells and by restitutive forces of the elastic membrane.

On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs

We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18].