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Parameterized Valiant’s Classes

Authors Markus Bläser, Christian Engels

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Author Details

Markus Bläser
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Christian Engels
  • IIT Bombay, Mumbai, India

Acknowledgements

We thank Holger Dell for valuable comments on a first draft of this paper and Radu Curticapean and Marc Roth for helpful discussions.

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Markus Bläser and Christian Engels. Parameterized Valiant’s Classes. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.IPEC.2019.3

Abstract

We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate reductions and completeness notions. Our main contribution is the VW[1]-completeness proof of the parameterized clique family. This proof is far more complicated than in the Boolean world. It requires some new concepts like composition theorems for bounded exponential sums and Boolean-arithmetic formulas. In addition, we also look at two polynomials linked to the permanent with vastly different parameterized complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic complexity theory
  • parameterized complexity theory
  • Valiant’s classes

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