Contributions to exact approaches in combinatorial optimization

The focus of this thesis is to contribute to the development of new, exact solution approaches to different combinatorial optimization problems. In particular, we derive dedicated algorithms for a special class of Traveling Tournament Problems (TTPs), the Dial-A-Ride Problem (DARP), and the Vehicle Routing Problem with Time Windows and Temporal Synchronized Pickup and Delivery (VRPTWTSPD). Furthermore, we extend the concept of using dual-optimal inequalities for stabilized Column Generation (CG) and detail its application to improved CG algorithms for the cutting stock problem, the bin packing problem, the vertex coloring problem, and the bin packing problem with conflicts. In all approaches, we make use of some knowledge about the structure of the problem at hand to individualize and enhance existing algorithms. Specifically, we utilize knowledge about the input data (TTP), problem-specific constraints (DARP and VRPTWTSPD), and the dual solution space (stabilized CG). Extensive computational results proving the usefulness of the proposed methods are reported.

Efficient certification of feasibility and objective value of linear programs and its applications

The use of linear programming in various areas has increased with the significant improvement of specialized solvers. Linear programs are used as such to model practical problems, or as subroutines in algorithms such as formal proofs or branch-and-cut frameworks. In many situations a certified answer is needed, for example the guarantee that the linear program is feasible or infeasible, or a provably safe bound on its objective value. Most of the available solvers work with floating-point arithmetic and are thus subject to its shortcomings such as rounding errors or underflow, therefore they can deliver incorrect answers. While adequate for some applications, this is unacceptable for critical applications like flight controlling or nuclear plant management due to the potential catastrophic consequences. We propose a method that gives a certified answer whether a linear program is feasible or infeasible, or returns unknown'. The advantage of our method is that it is reasonably fast and rarely answers unknown'. It works by computing a safe solution that is in some way the best possible in the relative interior of the feasible set. To certify the relative interior, we employ exact arithmetic, whose use is nevertheless limited in general to critical places, allowing us to rnremain computationally efficient. Moreover, when certain conditions are fulfilled, our method is able to deliver a provable bound on the objective value of the linear program. We test our algorithm on typical benchmark sets and obtain higher rates of success compared to previous approaches for this problem, while keeping the running times acceptably small. The computed objective value bounds are in most of the cases very close to the known exact objective values. We prove the usability of the method we developed by additionally employing a variant of it in a different scenario, namely to improve the results of a Satisfiability Modulo Theories solver. Our method is used as a black box in the nodes of a branch-and-bound tree to implement conflict learning based on the certificate of infeasibility for linear programs consisting of subsets of linear constraints. The generated conflict clauses are in general small and give good rnprospects for reducing the search space. Compared to other methods we obtain significant improvements in the running time, especially on the large instances.

Search for a spin-dependent short-range force between nucleons with a 3 He, 129 Xe clock-comparison experiment

Light pseudoscalar bosons, such as the axion that was originally proposed as a solution of the strong CP problem, would cause a new spin-dependent short-range interaction. In this thesis, an experiment is presented to search for axion mediated short-range interaction between a nucleon and the spin of a polarized bound neutron. This interaction cause a shift in the precession frequency of nuclear spin-polarized gases in the presence of an unpolarized mass. To get rid of magnetic field drifts co-located, nuclear spin polarized 3He and 129Xe atoms were used. The free nuclear spin precession frequencies were measured in a homogeneous magnetic guiding field of about 350nT using LTc SQUID detectors. The whole setup was housed in a magnetically shielded room at the Physikalisch Technische Bundesanstalt (PTB) in Berlin. With this setup long nuclear spin-coherence times, respectively, transverse relaxation times of 5h for 129Xe and 53h for 3He could be achieved. The results of the last run in September 2010 are presented which give new upper limits on the scalar-pseudoscalar coupling of axion-like particles in the axion-mass window from 10^(-2) eV to 10^(-6) eV. The laboratory upper bounds were improved by up to 4 orders of magnitude.

Analysis and numerical solution of the Peterlin viscoelastic model

Liquids and gasses form a vital part of nature. Many of these are complex fluids with non-Newtonian behaviour. We introduce a mathematical model describing the unsteady motion of an incompressible polymeric fluid. Each polymer molecule is treated as two beads connected by a spring. For the nonlinear spring force it is not possible to obtain a closed system of equations, unless we approximate the force law. The Peterlin approximation replaces the length of the spring by the length of the average spring. Consequently, the macroscopic dumbbell-based model for dilute polymer solutions is obtained. The model consists of the conservation of mass and momentum and time evolution of the symmetric positive definite conformation tensor, where the diffusive effects are taken into account. In two space dimensions we prove global in time existence of weak solutions. Assuming more regular data we show higher regularity and consequently uniqueness of the weak solution. For the Oseen-type Peterlin model we propose a linear pressure-stabilized characteristics finite element scheme. We derive the corresponding error estimates and we prove, for linear finite elements, the optimal first order accuracy. Theoretical error of the pressure-stabilized characteristic finite element scheme is confirmed by a series of numerical experiments.