A generalized method for proving polynomial calculus degree lower bounds
(2015) 30th Conference on Computational Complexity, CCC 2015 In Leibniz International Proceedings in Informatics, LIPIcs 33. p.467-487- Abstract
We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov'03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas... (More)
We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov'03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov'02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis'93], both FPHP and Onto-PHP formulas are hard even when restricted to bounded-degree expanders.
(Less)
- author
- Mikša, Mladen and Nordström, Jakob LU
- publishing date
- 2015-06-01
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- keywords
- Degree, Functional pigeonhole principle, Lower bound, PCR, Polynomial calculus, Polynomial calculus resolution, Proof complexity, Size
- host publication
- 30th Conference on Computational Complexity, CCC 2015
- series title
- Leibniz International Proceedings in Informatics, LIPIcs
- editor
- Zuckerman, David
- volume
- 33
- pages
- 21 pages
- publisher
- Schloss Dagstuhl - Leibniz-Zentrum für Informatik
- conference name
- 30th Conference on Computational Complexity, CCC 2015
- conference location
- Portland, United States
- conference dates
- 2015-06-17 - 2015-06-19
- external identifiers
-
- scopus:84958256296
- ISSN
- 1868-8969
- ISBN
- 9783939897811
- DOI
- 10.4230/LIPIcs.CCC.2015.467
- language
- English
- LU publication?
- no
- id
- 3fbb654b-fa15-4aa8-940f-9ce1af87d91e
- date added to LUP
- 2020-12-18 22:23:04
- date last changed
- 2022-04-26 22:58:01
@inproceedings{3fbb654b-fa15-4aa8-940f-9ce1af87d91e,
abstract = {{<p>We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov'03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov'02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis'93], both FPHP and Onto-PHP formulas are hard even when restricted to bounded-degree expanders.</p>}},
author = {{Mikša, Mladen and Nordström, Jakob}},
booktitle = {{30th Conference on Computational Complexity, CCC 2015}},
editor = {{Zuckerman, David}},
isbn = {{9783939897811}},
issn = {{1868-8969}},
keywords = {{Degree; Functional pigeonhole principle; Lower bound; PCR; Polynomial calculus; Polynomial calculus resolution; Proof complexity; Size}},
language = {{eng}},
month = {{06}},
pages = {{467--487}},
publisher = {{Schloss Dagstuhl - Leibniz-Zentrum für Informatik}},
series = {{Leibniz International Proceedings in Informatics, LIPIcs}},
title = {{A generalized method for proving polynomial calculus degree lower bounds}},
url = {{http://dx.doi.org/10.4230/LIPIcs.CCC.2015.467}},
doi = {{10.4230/LIPIcs.CCC.2015.467}},
volume = {{33}},
year = {{2015}},
}