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 nontrivial 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.

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Leibniz International Proceedings in Informatics
International Symposium on Parameterized and Exact Computation
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Bottesch, R.C. (2017). Relativization and interactive proof systems in Parameterized Complexity theory. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017) (pp. 9:1–9:12). doi:10.4230/LIPIcs.IPEC.2017.9