Total Space in Resolution Is at Least Width Squared

Given an unsatisfiable k-CNF formula phi we consider two complexity measures in Resolution: width and total space. The width is the minimal W such that there exists a Resolution refutation of phi with clauses of at most W literals. The total space is the minimal size T of a memory used to write down a Resolution refutation of phi where the size of the memory is measured as the total number of literals it can contain. We prove that T = Omega((W - k)^2)

Parameterized Approaches for Large-Scale Optimization Problems

In this dissertation, we study challenging discrete optimization problems from the perspective of parameterized complexity. The usefulness of this type of analysis is twofold. First, it can lead to efficient algorithms for large-scale problem instances. Second, the analysis can provide a rigorous explanation for why challenging problems might appear relatively easy in practice. We illustrate the approach on several different problems, including: the maximum clique problem in sparse graphs; 0-1 programs with many conflicts; and the node-weighted Steiner tree problem with few terminal nodes. We also study polyhedral counterparts to fixed-parameter tractable algorithms. Specifically, we provide fixed-parameter tractable extended formulations for independent set in tree-like graphs and for cardinality-constrained vertex covers