Polynomial calculus for optimization

MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem. Weighted Polynomial Calculus is...

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Authors: Bonacina, Ilario|||0000-0002-5697-8070, Bonet Carbonell, M. Luisa|||0000-0003-1646-7177, Levy Díaz, Jordi
Format: article
Publication Date:2024
Country:España
Institution:Universitat Politècnica de Catalunya (UPC)
Repository:UPCommons. Portal del coneixement obert de la UPC
Language:English
OAI Identifier:oai:upcommons.upc.edu:2117/416840
Online Access:https://hdl.handle.net/2117/416840
https://dx.doi.org/10.1016/j.artint.2024.104208
Access Level:Open access
Keyword:MaxSAT
SAT
Proof systems
Polynomial calculus
Algebraic reasoning
Proof complexity
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de cossos i polinomis
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spelling Polynomial calculus for optimizationBonacina, Ilario|||0000-0002-5697-8070Bonet Carbonell, M. Luisa|||0000-0003-1646-7177Levy Díaz, JordiMaxSATSATProof systemsPolynomial calculusAlgebraic reasoningProof complexityÀrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de cossos i polinomisMaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem. Weighted Polynomial Calculus is a natural generalization of the systems MaxSAT-Resolution and weighted Resolution. Unlike such systems, weighted Polynomial Calculus manipulates polynomials with coefficients in a finite field and either weights in ℕ or ℤ. We show the soundness and completeness of weighted Polynomial Calculus via an algorithmic procedure. Weighted Polynomial Calculus, with weights in ℕ and coefficients in F2, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in ℤ, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.This work was supported by the grant numbers PID2019-109137GB-C21, PID2019-109137GB-C22, IJC2018-035334-I, PID2022138506NB-C21, and PID2022-138506NB-C22 funded by AEI. This work was partially done while Maria Luisa Bonet and Jordi Levy were visiting the Simons Institute for the Theory of Computing during the Spring of 2023.Peer Reviewed20242024-12-0120242024-10-31journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/416840https://dx.doi.org/10.1016/j.artint.2024.104208reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2022-138506NB-C22 SATISFACTIBILIDAD Y OPTIMIZACION CON CERTIFICADOS DE PRUEBA MAS ALLA DE RESOLUCIONopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial 4.0 Internationalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4168402026-05-27T15:37:01Z
dc.title.none.fl_str_mv Polynomial calculus for optimization
title Polynomial calculus for optimization
spellingShingle Polynomial calculus for optimization
Bonacina, Ilario|||0000-0002-5697-8070
MaxSAT
SAT
Proof systems
Polynomial calculus
Algebraic reasoning
Proof complexity
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de cossos i polinomis
title_short Polynomial calculus for optimization
title_full Polynomial calculus for optimization
title_fullStr Polynomial calculus for optimization
title_full_unstemmed Polynomial calculus for optimization
title_sort Polynomial calculus for optimization
dc.creator.none.fl_str_mv Bonacina, Ilario|||0000-0002-5697-8070
Bonet Carbonell, M. Luisa|||0000-0003-1646-7177
Levy Díaz, Jordi
author Bonacina, Ilario|||0000-0002-5697-8070
author_facet Bonacina, Ilario|||0000-0002-5697-8070
Bonet Carbonell, M. Luisa|||0000-0003-1646-7177
Levy Díaz, Jordi
author_role author
author2 Bonet Carbonell, M. Luisa|||0000-0003-1646-7177
Levy Díaz, Jordi
author2_role author
author
dc.subject.none.fl_str_mv MaxSAT
SAT
Proof systems
Polynomial calculus
Algebraic reasoning
Proof complexity
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de cossos i polinomis
topic MaxSAT
SAT
Proof systems
Polynomial calculus
Algebraic reasoning
Proof complexity
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de cossos i polinomis
description MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem. Weighted Polynomial Calculus is a natural generalization of the systems MaxSAT-Resolution and weighted Resolution. Unlike such systems, weighted Polynomial Calculus manipulates polynomials with coefficients in a finite field and either weights in ℕ or ℤ. We show the soundness and completeness of weighted Polynomial Calculus via an algorithmic procedure. Weighted Polynomial Calculus, with weights in ℕ and coefficients in F2, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in ℤ, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.
publishDate 2024
dc.date.none.fl_str_mv 2024
2024-12-01
2024
2024-10-31
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/416840
https://dx.doi.org/10.1016/j.artint.2024.104208
url https://hdl.handle.net/2117/416840
https://dx.doi.org/10.1016/j.artint.2024.104208
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2022-138506NB-C22 SATISFACTIBILIDAD Y OPTIMIZACION CON CERTIFICADOS DE PRUEBA MAS ALLA DE RESOLUCION
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial 4.0 International
http://creativecommons.org/licenses/by-nc/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial 4.0 International
http://creativecommons.org/licenses/by-nc/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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