Lower Bounds for Heuristic Algorithms

The constraint satisfaction problem, the Boolean satisfiability problem and the graph isomorphism problem do not have efficient algorithms. In order to solve these problems, one utilizes heuristic algorithms of polynomial running time. The present thesis studies three classical heuristics for the above-mentioned decision problems and answers the question whether they can be implemented more efficient than with the fastest known algorithms. The k-consistency heuristic for the constraint satisfaction problem tries to establish local consistency by iteratively propagating new constraints and can be implemented time O(n2k). We show that the degree of the polynomial that bounds the running time has to increase linear in k. To achieve this, we prove for a fixed constant c > 0 that there is no algorithm of running time O(nck) that decides whether k-consistency can be established. Furthermore, we analyze the propagation process of k-consistency algorithms and prove optimal lower bounds on the number of nested propagation steps. One heuristic for the Boolean satisfiability problem is to find resolution refutations in which every clause contains at most k literals. Such refutations of width k can be found in time O(nk+1) by a simple search procedure. We show for a fixed constant c > 0, that it cannot be decided in time O(nck) whether a given formula has a resolution refutation of width k. The lower bounds on the time complexity for k-consistency and bounded width resolution do not rely on complexity theoretic assumptions and demonstrate one of the rare examples where the deterministic time hierarchy theorem can be applied to natural decision problems. The color refinement heuristic for the graph isomorphism problem computes a stable coloring of the vertices of a graph by iteratively refining the color classes. By refining the color classes in a clever order, the color refinement procedure can be implemented in time O(m log n) on connected graphs with n vertices and m edges. We show that this refining strategy is optimal. To prove this, we construct graphs where every possible order of refining operations needs at least Ω(m log n) computation steps.