Application of the Branch and Cut Method to the Vehicle Routing Problem

The successful application of Branch and Cut methods to the TSP has drawn attention also to the polyhedral properties of the symmetric capacitated vehicle routing problem, which is the capacitated counterpart of the TSP. We investigate three classes of valid inequalities for the CVRP, multistars, pathbin inequalities and hypotours and give computational results we obtained with a Branch and Cut implementation

The critical exponents of the two-dimensional Ising spin glass revisited: Exact Ground State Calculations and Monte Carlo Simulations

The critical exponents for T -> 0 of the two-dimensional Ising spin glass model with Gaussian couplings are determined with the help of exact ground states for system sizes up to L=50 and by a Monte Carlo study of a pseudo-ferromagnetic order parameter. We obtain: for the stiffness exponent y(= heta)=-0.281 ±0.002 , for the magnetic exponent delta=1.48 ±0.01 and for the chaos exponent zeta=1.05 ±0.05 . From Monte Carlo simulations we get the thermal exponent u=3.6 ±0.2 . The scaling prediction y=-1/u is fulfilled within the error bars, whereas there is a disagreement with the relation y=1-delta

Ground state properties of solid-on-solid models with disordered substrates

We study the glassy super-rough phase of a class of solid-on-solid models with a disordered substrate in the limit of vanishing temperature by means of exact ground states, which we determine with a newly developed minimum cost flow algorithm. Results for the height-height correlation function are compared with analytical and numerical predictions. The domain wall energy of a boundary induced step grows logarithmically with system size, indicating the marginal stability of the ground state, and the fractal dimension of the step is estimated. The sensibility of the ground state with respect to infinitesimal variations of the quenched disorder is analyzed.Comment: 4 pages RevTeX, 3 eps-figures include

Anwendung des Branch&Cut Verfahrens auf das kapazitierte Vehicle Routing Problem

The successful application of Branch and Cut methods to the TSP has drawn attention also to the polyhedral properties of the symmetric capacitated vehicle routing problem, which is the capacitated counterpart of the TSP. In this PhD-theses several classes of valid inequalities for the CVRP, like multistars, pathbin inequalities and hypotours are investigated. We also give computational results we obtained with a Branch and Cut implementation. We were able to solve two of the difficult 76-node benchmarks from Eilon to optimality

Application of the branch and cut method to the vehicle routing problem

The successful application of Branch and Cut methods to the TSP has drawn attention also to the polyhedral properties of the symmetric capacitated vehicle routing problem, which is the capacitated counterpart of the TSP. We investigate three classes of valid inequalities for the CVRP, multistars, pathbin inequalities and hypotours and give computational results we obtained with a Branch and Cut implementation

Steiner-Diagrams

In this report, we introduce and study the Steiner-diagram-problem. Given a transitive digraph G = (V; A; w) with non-negative edge-weights and a set of demand edges B, the objective is to find an acyclic set of edges S of minimal cost, whose transitive closure contains B. This problem is NP-complete in the general case and has some interesting structural properties that make it polynomially solvable if the size of B is bounded by a constant, the triangle inequality holds in A and A is transitively closed

Steiner-Diagrams and k-Star-Hubs

In this report, we introduce and study two problems derived from reload problems in transport logistics. Given a transitive digraph G=(V,A,w) with non-negative edge-weights and a set of demand edges B, the objective of the Steiner-diagram-problem is to find an acyclic set of edges S of minimal cost, whose transitive closure contains B. This problem is NP-complete in the general case and has some interesting structural properties that make it polynomially solvable if the size of B is bounded by a constant, the triangle inequality holds in A and A is transitively closed. Secondly, we discuss a weighted edge-cover problem with k cost functions on the vertices and give an efficient algorithm for the case k = 2. This report is an extended version of 99-342

Scheduling Trams in the Morning

In this note, we prove NP-completeness of the following problem: Given a set of trams of different types, which are stacked on sidings in their depot and an order in which trams of specified types are supposed to leave. Is there an assignment of trams to departure times without any shunting movements? In the particular case where the number of sidings is fixed, the problem is solvable in polynomial time. We derive a dynamic program and improve its performance by a state elimination scheme. We implemented three variants of the dynamic program and applied them to random data as well as to real-world data

Scheduling Trams in the Morning is Hard

In this note, we prove NP-completeness of the following problem: Given a set of trams of different types, which are stacked on sidings in their depot and an order in which trams of specified types are supposed to leave. Is there an assignment of trams to departure times without any shunting movements? For the special case where the number of sidings is fixed the problem is solvable in polynomial time. We derive a brute force and a more sophisticated implementation of the associated algorithm. Furthermore, we compare the implementations on random and real word data