NDLI: Parameterized Proof Complexity

Content Provider	IEEE Xplore Digital Library
Author	Dantchev, S. Martin, B. Szeider, S.
Copyright Year	2007
Description	Author affiliation: Durham Univ., Durham (Dantchev, S.; Martin, B.; Szeider, S.)
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 CNF 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(M. Cesati, 2006) by showing that there is no fpt-bounded parameterized proof system, 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 sub-cases, 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.
Starting Page	150
Ending Page	160
File Size	251916
Page Count	11
File Format	PDF
ISBN	9780769530109
ISSN	02725428
DOI	10.1109/FOCS.2007.53
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2007-10-21
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Polynomials Turing machines Computer science Algorithm design and analysis NP-hard problem Design engineering Councils
Content Type	Text
Resource Type	Article

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