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DOI: 10.4230/DagSemProc.10061.4
URN: urn:nbn:de:0030-drops-25254
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2525/
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Beyersdorff, Olaf ;
Galesi, Nicola ;
Lauria, Massimo
Hardness of Parameterized Resolution
Abstract
Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles.
We broadly investigate Parameterized Resolution obtaining the following main results:
1) We introduce a purely combinatorial approach to obtain lower bounds to the proof size in tree-like Parameterized Resolution. For this we devise a new asymmetric Prover-Delayer game which characterizes proofs in (parameterized) tree-like Resolution.
By exhibiting good Delayer strategies we then show lower bounds for the pigeonhole principle as well as the order principle.
2) Interpreting a well-known FPT algorithm for vertex cover as a DPLL procedure for Parameterized Resolution, we devise a proof search algorithm for Parameterized Resolution and show that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's.
3) We answer a question posed by Dantchev, Martin, and Szeider showing that dag-like Parameterized Resolution is not fpt-bounded. We obtain this result by proving that the pigeonhole principle requires proofs of size $n^{Omega(k)}$ in dag-like Parameterized Resolution.
For this lower bound we use a different Prover-Delayer game which was developed for Resolution by Pudlák.
BibTeX - Entry
@InProceedings{beyersdorff_et_al:DagSemProc.10061.4,
author = {Beyersdorff, Olaf and Galesi, Nicola and Lauria, Massimo},
title = {{Hardness of Parameterized Resolution}},
booktitle = {Circuits, Logic, and Games},
pages = {1--28},
series = {Dagstuhl Seminar Proceedings (DagSemProc)},
ISSN = {1862-4405},
year = {2010},
volume = {10061},
editor = {Benjamin Rossman and Thomas Schwentick and Denis Th\'{e}rien and Heribert Vollmer},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2010/2525},
URN = {urn:nbn:de:0030-drops-25254},
doi = {10.4230/DagSemProc.10061.4},
annote = {Keywords: Proof complexity, parameterized complexity, Resolution, Prover-Delayer Games}
}
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Keywords: |
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Proof complexity, parameterized complexity, Resolution, Prover-Delayer Games |
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Collection: |
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10061 - Circuits, Logic, and Games |
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Issue Date: |
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2010 |
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Date of publication: |
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26.04.2010 |
