On the power of symmetric linear programs

We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corres...

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Detalles Bibliográficos
Autores: Atserias, Albert|||0000-0002-3732-1989, Dawar, Anuj, Ochremiak, Joanna Regina
Tipo de recurso: artículo
Fecha de publicación:2021
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/384806
Acceso en línea:https://hdl.handle.net/2117/384806
https://dx.doi.org/10.1145/3456297
Access Level:acceso abierto
Palabra clave:Linear programming
Extended formulations
Planted clique
Perfect matching
Symmetric group
Programació lineal
Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica
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spelling On the power of symmetric linear programsAtserias, Albert|||0000-0002-3732-1989Dawar, AnujOchremiak, Joanna ReginaLinear programmingExtended formulationsPlanted cliquePerfect matchingSymmetric groupProgramació linealÀrees temàtiques de la UPC::Informàtica::Informàtica teòricaWe consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs, while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991), showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.First author partially funded by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement ERC-2014-CoG 648276 (AUTAR) and MICCIN grant TIN2016-76573-C2-1P (TASSAT3). The second author was partially supported by a Fellowship of the Alan Turing Institute under the EPSRC grant EP/N510129/1. The third author funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 795936.Peer Reviewed20212021-08-0120232023-03-09journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/384806https://dx.doi.org/10.1145/3456297reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengEuropean Commission http://doi.org/10.13039/100010661 Horizon 2020 Framework Programme 648276 A Unified Theory of Algorithmic Relaxationsopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/3848062026-05-27T15:37:01Z
dc.title.none.fl_str_mv On the power of symmetric linear programs
title On the power of symmetric linear programs
spellingShingle On the power of symmetric linear programs
Atserias, Albert|||0000-0002-3732-1989
Linear programming
Extended formulations
Planted clique
Perfect matching
Symmetric group
Programació lineal
Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica
title_short On the power of symmetric linear programs
title_full On the power of symmetric linear programs
title_fullStr On the power of symmetric linear programs
title_full_unstemmed On the power of symmetric linear programs
title_sort On the power of symmetric linear programs
dc.creator.none.fl_str_mv Atserias, Albert|||0000-0002-3732-1989
Dawar, Anuj
Ochremiak, Joanna Regina
author Atserias, Albert|||0000-0002-3732-1989
author_facet Atserias, Albert|||0000-0002-3732-1989
Dawar, Anuj
Ochremiak, Joanna Regina
author_role author
author2 Dawar, Anuj
Ochremiak, Joanna Regina
author2_role author
author
dc.subject.none.fl_str_mv Linear programming
Extended formulations
Planted clique
Perfect matching
Symmetric group
Programació lineal
Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica
topic Linear programming
Extended formulations
Planted clique
Perfect matching
Symmetric group
Programació lineal
Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica
description We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs, while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991), showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.
publishDate 2021
dc.date.none.fl_str_mv 2021
2021-08-01
2023
2023-03-09
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/384806
https://dx.doi.org/10.1145/3456297
url https://hdl.handle.net/2117/384806
https://dx.doi.org/10.1145/3456297
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv European Commission http://doi.org/10.13039/100010661 Horizon 2020 Framework Programme 648276 A Unified Theory of Algorithmic Relaxations
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
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
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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