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Exponential resolution lower bounds for weak pigeonhole principle and perfect matching formulas over sparse graphs

de Rezende, Susanna F. LU orcid ; Nordström, Jakob LU ; Risse, Kilian and Sokolov, Dmitry LU (2020) 35th Computational Complexity Conference, CCC 2020 In Leibniz International Proceedings in Informatics, LIPIcs 169.
Abstract

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson'01] and highly unbalanced, dense graphs as in [Raz'04] and [Razborov'03,'04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof... (More)

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson'01] and highly unbalanced, dense graphs as in [Raz'04] and [Razborov'03,'04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.

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Please use this url to cite or link to this publication:
author
; ; and
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Perfect matching, Proof complexity, Resolution, Sparse graphs, Weak pigeonhole principle
host publication
CCC '20: Proceedings of the 35th Computational Complexity Conference 2020
series title
Leibniz International Proceedings in Informatics, LIPIcs
editor
Saraf, Shubhangi
volume
169
article number
28
publisher
Schloss Dagstuhl - Leibniz-Zentrum für Informatik
conference name
35th Computational Complexity Conference, CCC 2020
conference location
Virtual, Online, Germany
conference dates
2020-07-28 - 2020-07-31
external identifiers
  • scopus:85089398484
ISSN
1868-8969
ISBN
9783959771566
DOI
10.4230/LIPIcs.CCC.2020.28
language
English
LU publication?
yes
id
1529fede-06f1-4730-ac52-e0a29cd502f2
date added to LUP
2020-12-18 22:14:03
date last changed
2022-04-19 02:56:18
@inproceedings{1529fede-06f1-4730-ac52-e0a29cd502f2,
  abstract     = {{<p>We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson'01] and highly unbalanced, dense graphs as in [Raz'04] and [Razborov'03,'04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.</p>}},
  author       = {{de Rezende, Susanna F. and Nordström, Jakob and Risse, Kilian and Sokolov, Dmitry}},
  booktitle    = {{CCC '20: Proceedings of the 35th Computational Complexity Conference 2020}},
  editor       = {{Saraf, Shubhangi}},
  isbn         = {{9783959771566}},
  issn         = {{1868-8969}},
  keywords     = {{Perfect matching; Proof complexity; Resolution; Sparse graphs; Weak pigeonhole principle}},
  language     = {{eng}},
  month        = {{07}},
  publisher    = {{Schloss Dagstuhl - Leibniz-Zentrum für Informatik}},
  series       = {{Leibniz International Proceedings in Informatics, LIPIcs}},
  title        = {{Exponential resolution lower bounds for weak pigeonhole principle and perfect matching formulas over sparse graphs}},
  url          = {{http://dx.doi.org/10.4230/LIPIcs.CCC.2020.28}},
  doi          = {{10.4230/LIPIcs.CCC.2020.28}},
  volume       = {{169}},
  year         = {{2020}},
}