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Probabilistic Parameterized Polynomial Time

  • Nils DonselaarEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

We examine a parameterized complexity class for randomized computation where only the error bound and not the full runtime is allowed to depend more than polynomially on the parameter, based on a proposal by Kwisthout in [15, 16]. We prove that this class, for which we propose the shorthand name PPPT, has a robust definition and is in fact equal to the intersection of the classes paraBPP and PP. This result is accompanied by a Cook-style proof of completeness for the corresponding promise class (under a suitable notion of reduction) for parameterized approximation versions of the inference problem in Bayesian networks, which is known to be PP-complete. With these definitions and results in place, we proceed by showing how it follows from this that derandomization is equivalent to efficient deterministic approximation methods for the inference problem. Furthermore, we observe as a straightforward application of a result due to Drucker in [8] that these problems cannot have polynomial size randomized kernels unless the polynomial hierarchy collapses to the third level. We conclude by indicating potential avenues for further exploration and application of this framework.

Keywords

Parameterized complexity theory Randomized computation Bayesian networks 

Notes

Acknowledgement

The author thanks Johan Kwisthout and Hans Bodlaender for sharing insightful remarks in his discussions with them, and also Ralph Bottesch for providing useful comments on an early draft of this paper.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Donders Institute for Brain, Cognition and BehaviourRadboud UniversityNijmegenThe Netherlands

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