## 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.

## Keywords

Propositional proof complexity parameterized complexity complexity gap theorems## Subject classification

68Q25## References

- Samuel Buss & Tonian Pitassi (1998). Resolution and the weak pigeonhole principle. In:
*CSL ’97, LNCS 1414*, 149–156. Springer Verlag.Google Scholar - Marco Cesati (2006). Compendium of Parameterized problems. http://bravo.ce.uniroma2.it/home/cesati/research/compendium/.
- Jianer Chen, Iyad A. Kanj & Ge Xia (2006). Improved Parameterized Upper Bounds for Vertex Cover. In:
*Proceedings of MFCS 2006, LNCS 4162*, 238–249. Springer Verlag.Google Scholar - Stephen Cook, Robert Reckhow (1979) The relative efficiency of propositional proof systems. J. Symbolic Logic 44(1): 36–50MathSciNetzbMATHCrossRefGoogle Scholar
- Rodney G. Downey & Michael R. Fellows (1999).
*Parameterized Complexity*. Monographs in Computer Science. Springer Verlag.Google Scholar - Jörg Flum & Martin Grohe (2006).
*Parameterized Complexity Theory*, volume XIV of*Texts in Theoretical Computer Science. An EATCS Series*. Springer Verlag.Google Scholar - Russell Impagliazzo, Ramamohan Paturi, Francis Zane (2001) Which problems have strongly exponential complexity. J. of Computer and System Sciences 63(4): 512–530zbMATHCrossRefGoogle Scholar
- Jan Krajíček (1995).
*Bounded arithmetic, propositional logic, and complexity theory*. Cambridge University Press, New York, NY, USA. ISBN 0-521-45205-8.Google Scholar - Rolf Niedermeier (2006).
*Invitation to Fixed-Parameter Algorithms*. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press.Google Scholar - Pavel Pudlák (2000) Proofs as Games. American Mathematical Monthly 107(6): 541–550MathSciNetzbMATHCrossRefGoogle Scholar
- Soren Riis (2001) A complexity gap for tree-resolution. Computational Complexity 3(10): 179–209Google Scholar