The quantified Boolean formula (QBF) problem is an important decision problem generally viewed as the archetype for PSPACE-completeness. Many problems of central interest in AI are in general not included in NP, e.g., planning, model checking, and non-monotonic reasoning, and for such problems QBF has successfully been used as a modelling tool. However, solvers for QBF are not as advanced as state of the art SAT solvers, which has prevented QBF from becoming a universal modelling language for PSPACE-complete problems. A theoretical explanation is that QBF (as well as many other PSPACE-complete problems) lacks natural parameters} guaranteeing fixed-parameter tractability (FPT). In this paper we tackle this problem and consider a simple but overlooked parameter: the number of existentially quantified variables. This natural parameter is virtually unexplored in the literature which one might find surprising given the general scarcity of FPT algorithms for QBF. Via this parameterization we then develop a novel FPT algorithm applicable to QBF instances in conjunctive normal form (CNF) of bounded clause length. We complement this by a W[1]-hardness result for QBF in CNF of unbounded clause length as well as sharper lower bounds for the bounded arity case under the (strong) exponential-time hypothesis.

In this paper we consider the problem of deciding if a (likely incomplete) description of a system of events is consistent, the network consistency problem for the point algebra of partially ordered time (POT). This is achieved by a sophisticated enumeration of structures similar to total orders, which are then greedily expanded towards a solution. While similar ideas have been explored earlier for related problems it turns out that the analysis for POT is non-trivial and requires significant new ideas. Qualitative reasoning is an important formalism in artificial intelligence where the objective is to reason about continuous properties given certain relations between the unknown entities. Two important subfields are temporal reasoning, e.g., the point algebra for partially ordered time (POT), Allen's interval algebra (A), and the point algebra for branching time, and spatial reasoning, e.g., the region connection calculus (RCC), the cardinal direction calculus, and the rectangle algebra.

Allen's interval algebra is one of the most well-known calculi in qualitative temporal reasoning with numerous applications in artificial intelligence. Recently, there has been a surge of improvements in the fine-grained complexity of NP-hard reasoning tasks, improving the running time from the naive $2^{O(n^2)}$ to $O^*((1.0615n)^{n})$, with even faster algorithms for unit intervals a bounded number of overlapping intervals (the $O^*(\cdot)$ notation suppresses polynomial factors). Despite these improvements the best known lower bound is still only $2^{o(n)}$ (under the exponential-time hypothesis) and major improvements in either direction seemingly require fundamental advances in computational complexity. In this paper we propose a novel framework for solving NP-hard qualitative reasoning problems which we refer to as dynamic programming with sublinear partitioning. Using this technique we obtain a major improvement of $O^*((\frac{cn}{\log{n}})^{n})$ for Allen's interval algebra. To demonstrate that the technique is applicable to more domains we apply it to a problem in qualitative spatial reasoning, the cardinal direction point algebra, and solve it in $O^*((\frac{cn}{\log{n}})^{2n/3})$ time. Hence, not only do we significantly advance the state-of-the-art for NP-hard qualitative reasoning problems, but obtain a novel algorithmic technique that is likely applicable to many problems where $2^{O(n)}$ time algorithms are unlikely.