Grundzustandsstruktur ungeordneter Systeme und Dynamik von Optimierungsalgorithmen

Abstract

Combinatorial optimization problems from Theoretical Computer Sciences and certain magnetic alloys (called spin glasses) both consist of particles or variables which are coupled by a large number of competing interactions. This thesis deals with three such systems, for which determining an optimal state (also called ground state or solution) in which a maximal number of interaction is satisfied, is an algorithmically difficult problem.The difference between the ground state and the first excited state of 3-d Ising spin-glasses is studied using a substantially improved algorithm for finding ground states. The probability of a system wide excitation does not decrease with system size, in agreement with the predictions made by the mean-field model.The ground-state structure of the vertex-cover problem from Theoretical Computer Sciences is analyzed for a certain ensemble of random graphs using numerical clustering methods originally developed also for spin glasses. In a certain region of the characterizing parameter the ground state landscape consists only of a single cluster, which is in agreement with previously known analytical results. In the other region ground-states are organized in a hierarchical way which is compatible with continuous replica symmetry-breaking.For the random satisfiability problem the dynamics of an algorithm for finding solutions is described analytically. Two phases are identified: while solutions are found fast, i.e. in a time growing linear with system size, in one region, the solution time grows exponentially in the other. Qualitatively, the analytically predicted solution times agree with numerical simulations.