Relativization and Interactive Proof Systems in Parameterized Complexity Theory

We introduce some classical complexity-theoretic techniques to Parameterized Complexity. First, we study relativization for the machine models that were used by Chen, Flum, and Grohe (2005) to characterize a number of parameterized complexity classes. Here we obtain a new and non-trivial characterization of the A-Hierarchy in terms of oracle machines, and parameterize a famous result of Baker, Gill, and Solovay (1975), by proving that, relative to specific oracles, FPT and A[1] can either coincide or differ (a similar statement holds for FPT and W[P]). Second, we initiate the study of interactive proof systems in the parameterized setting, and show that every problem in the class AW[SAT] has a proof system with "short" interactions, in the sense that the number of rounds is upper-bounded in terms of the parameter value alone

Levelable Sets and the Algebraic Structure of Parameterizations

Asking which sets are fixed-parameter tractable for a given parameterization constitutes much of the current research in parameterized complexity theory. This approach faces some of the core difficulties in complexity theory. By focussing instead on the parameterizations that make a given set fixed-parameter tractable, we circumvent these difficulties. We isolate parameterizations as independent measures of complexity and study their underlying algebraic structure. Thus we are able to compare parameterizations, which establishes a hierarchy of complexity that is much stronger than that present in typical parameterized algorithms races. Among other results, we find that no practically fixed-parameter tractable sets have optimal parameterizations

On the Average-case Complexity of Parameterized Clique

The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its average-case complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the average-case complexity of the k-Clique problem on the parameter k. To this end, we define two natural parameterized analogs of efficient average-case algorithms. We then show that k-Clique admits both analogues for Erd\H{o}s-R\'{e}nyi random graphs of arbitrary density. We also show that k-Clique is unlikely to admit neither of these analogs for some specific computable input distribution

Towards Work-Efficient Parallel Parameterized Algorithms

Parallel parameterized complexity theory studies how fixed-parameter tractable (fpt) problems can be solved in parallel. Previous theoretical work focused on parallel algorithms that are very fast in principle, but did not take into account that when we only have a small number of processors (between 2 and, say, 1024), it is more important that the parallel algorithms are work-efficient. In the present paper we investigate how work-efficient fpt algorithms can be designed. We review standard methods from fpt theory, like kernelization, search trees, and interleaving, and prove trade-offs for them between work efficiency and runtime improvements. This results in a toolbox for developing work-efficient parallel fpt algorithms.Comment: Prior full version of the paper that will appear in Proceedings of the 13th International Conference and Workshops on Algorithms and Computation (WALCOM 2019), February 27 - March 02, 2019, Guwahati, India. The final authenticated version is available online at https://doi.org/10.1007/978-3-030-10564-8_2

Model-Checking Problems as a Basis for Parameterized Intractability

Most parameterized complexity classes are defined in terms of a parameterized version of the Boolean satisfiability problem (the so-called weighted satisfiability problem). For example, Downey and Fellow's W-hierarchy is of this form. But there are also classes, for example, the A-hierarchy, that are more naturally characterised in terms of model-checking problems for certain fragments of first-order logic. Downey, Fellows, and Regan were the first to establish a connection between the two formalisms by giving a characterisation of the W-hierarchy in terms of first-order model-checking problems. We improve their result and then prove a similar correspondence between weighted satisfiability and model-checking problems for the A-hierarchy and the W^*-hierarchy. Thus we obtain very uniform characterisations of many of the most important parameterized complexity classes in both formalisms. Our results can be used to give new, simple proofs of some of the core results of structural parameterized complexity theory.Comment: Changes in since v2: Metadata update