Topologically correct intersection curves of tori and natural quadrics

This thesis provides efficient and robust algorithms for the computation of the intersection curve between a torus and a simple surface (e.g. a plane, a natural quadric or another torus), based on algebraic and numeric methods. The algebraic part includes the classification of the topological type of the intersection curve and the detection of degenerate situations like embedded conic sections and singularities. Moreover, reference points for each connected intersection curve component are determined. The required computations are realised efficiently by solving quartic polynomials at most and exactly by using exact arithmetic. The numeric part includes algorithms for the tracing of each intersection curve component, starting from the previously computed reference points. Using interval arithmetic, accidental incorrectness like jumping between branches or the skipping of parts are prevented. Furthermore, the environments of singularities are correctly treated. Our algorithms are complete in the sense that any kind of input can be handled including degenerate and singular configurations. They are verified, since the results are topologically correct and approximate the real intersection curve up to any arbitrary given error bound. The algorithms are robust, since no human intervention is required and they are efficient in the way that the treatment of algebraic equations of high degree is avoided.

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.