Parameterized proof complexity

Dantchev, Stefan, Martin, Barnaby and Szeider, Stefan (2011) Parameterized proof complexity. Computational Complexity, 20 (1). pp. 51-85. ISSN 1016-3328 (doi:10.1007/s00037-010-0001-1)

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Abstract

We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter (we call such a formula a parameterized contradiction). One could separate the parameterized complexity classes FPT and W[SAT] by showing that there is no fpt-bounded parameterized proof system for parameterized contradictions, i.e., that there is no proof system that admits proofs of size f(k)n O(1) where f is a computable function and n represents the size of the propositional formula. By way of a first step, we introduce the system of parameterized tree-like resolution and show that this system is not fpt-bounded. Indeed, we give a general result on the size of shortest tree-like resolution proofs of parameterized contradictions that uniformly encode first-order principles over a universe of size n. We establish a dichotomy theorem that splits the exponential case of Riis’s complexity gap theorem into two subcases, one that admits proofs of size f(k)n O(1) and one that does not. We also discuss how the set of parameterized contradictions may be embedded into the set of (ordinary) contradictions by the addition of new axioms. When embedded into general (DAG-like) resolution, we demonstrate that the pigeonhole principle has a proof of size 2 k n 2. This contrasts with the case of tree-like resolution where the embedded pigeonhole principle falls into the “non-FPT” category of our dichotomy.

Item Type: Article
Research Areas: A. > School of Science and Technology > Computer Science
A. > School of Science and Technology > Computer Science > Foundations of Computing group
Item ID: 9640
Depositing User: Devika Mohan
Date Deposited: 20 Dec 2012 10:38
Last Modified: 24 Apr 2018 14:26
URI: https://eprints.mdx.ac.uk/id/eprint/9640

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