The complexity of recognizing graph properties and the evasiveness conjecture
Deciding graph properties—Boolean functions invariant under graph isomorphism—is a foundational challenge in computational complexity. A central question is determining the minimum number of edge queries required to verify such properties in the worst case. A property is evasive if resolving it dema...
- Autores:
-
Mantilla Acosta, Rafael José
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2025
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/76183
- Acceso en línea:
- https://hdl.handle.net/1992/76183
- Palabra clave:
- Evasiveness Conjecture
Monotone graph properties
Deterministic query complexity
Topological methods in complexity
Matemáticas
- Rights
- openAccess
- License
- Attribution 4.0 International
| id |
UNIANDES2_bfce34da872a0a38b0ba07418929fe94 |
|---|---|
| oai_identifier_str |
oai:repositorio.uniandes.edu.co:1992/76183 |
| network_acronym_str |
UNIANDES2 |
| network_name_str |
Séneca: repositorio Uniandes |
| repository_id_str |
|
| dc.title.eng.fl_str_mv |
The complexity of recognizing graph properties and the evasiveness conjecture |
| dc.title.alternative.spa.fl_str_mv |
La complejidad de reconocer propiedades de grafos y la conjetura de evasividad |
| title |
The complexity of recognizing graph properties and the evasiveness conjecture |
| spellingShingle |
The complexity of recognizing graph properties and the evasiveness conjecture Evasiveness Conjecture Monotone graph properties Deterministic query complexity Topological methods in complexity Matemáticas |
| title_short |
The complexity of recognizing graph properties and the evasiveness conjecture |
| title_full |
The complexity of recognizing graph properties and the evasiveness conjecture |
| title_fullStr |
The complexity of recognizing graph properties and the evasiveness conjecture |
| title_full_unstemmed |
The complexity of recognizing graph properties and the evasiveness conjecture |
| title_sort |
The complexity of recognizing graph properties and the evasiveness conjecture |
| dc.creator.fl_str_mv |
Mantilla Acosta, Rafael José |
| dc.contributor.advisor.none.fl_str_mv |
Ángel Cárdenas, Jairo Andrés |
| dc.contributor.author.none.fl_str_mv |
Mantilla Acosta, Rafael José |
| dc.contributor.jury.none.fl_str_mv |
Bogart, Tristram Charles |
| dc.subject.keyword.eng.fl_str_mv |
Evasiveness Conjecture Monotone graph properties Deterministic query complexity Topological methods in complexity |
| topic |
Evasiveness Conjecture Monotone graph properties Deterministic query complexity Topological methods in complexity Matemáticas |
| dc.subject.themes.spa.fl_str_mv |
Matemáticas |
| description |
Deciding graph properties—Boolean functions invariant under graph isomorphism—is a foundational challenge in computational complexity. A central question is determining the minimum number of edge queries required to verify such properties in the worst case. A property is evasive if resolving it demands inspecting every potential edge. While most graph properties are non-evasive, the Evasiveness Conjecture posits that all monotone graph properties (those preserved under edge addition) are evasive. Though unresolved in general, this conjecture is proven for graphs with prime power vertex counts, a landmark result deeply tied to algebraic and topological methods. This work synthesizes advances in the Evasiveness Conjecture, providing a detailed exposition of its proof for prime power-sized graphs. Our analysis reveals unexpected connections to finite group theory and simplicial topology, underscoring how combinatorial problems of this nature straddle diverse mathematical domains. By unifying these perspectives, we clarify the conjecture’s current boundaries and the role of symmetry in computational hardness. |
| publishDate |
2025 |
| dc.date.accessioned.none.fl_str_mv |
2025-05-20T20:19:05Z |
| dc.date.available.none.fl_str_mv |
2025-05-20T20:19:05Z |
| dc.date.issued.none.fl_str_mv |
2025-04-20 |
| dc.type.none.fl_str_mv |
Trabajo de grado - Pregrado |
| dc.type.driver.none.fl_str_mv |
info:eu-repo/semantics/bachelorThesis |
| dc.type.version.none.fl_str_mv |
info:eu-repo/semantics/acceptedVersion |
| dc.type.coar.none.fl_str_mv |
http://purl.org/coar/resource_type/c_7a1f |
| dc.type.content.none.fl_str_mv |
Text |
| dc.type.redcol.none.fl_str_mv |
http://purl.org/redcol/resource_type/TP |
| format |
http://purl.org/coar/resource_type/c_7a1f |
| status_str |
acceptedVersion |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/1992/76183 |
| dc.identifier.instname.none.fl_str_mv |
instname:Universidad de los Andes |
| dc.identifier.reponame.none.fl_str_mv |
reponame:Repositorio Institucional Séneca |
| dc.identifier.repourl.none.fl_str_mv |
repourl:https://repositorio.uniandes.edu.co/ |
| url |
https://hdl.handle.net/1992/76183 |
| identifier_str_mv |
instname:Universidad de los Andes reponame:Repositorio Institucional Séneca repourl:https://repositorio.uniandes.edu.co/ |
| dc.language.iso.none.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.none.fl_str_mv |
Andrés Angel and Jerson Borja. Simplicial complexes and the evasiveness conjecture. Graduate Journal of Mathematics, 1(1), Jun 2016. Andres Angel and Jerson Borja. The evasiveness conjecture and graphs on 2p vertices. Journal of Graph Theory, 91, 05 2019. László Babai, Anandam Banerjee, Raghav Kulkarni, and Vipul Naik. Evasiveness and the distribution of prime numbers. CoRR, abs/1001.4829, 2010. Bruno Benedetti and Frank H. Lutz. The dunce hat in a minimal non-extendably collapsible 3-ball, 2009. Amit Chakrabarti, Subhash Khot, and Yaoyun Shi. Evasiveness of subgraph containment and related properties. In Afonso Ferreira and Horst Reichel, editors, STACS 2001, pages 110-120, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg. Sui-Xiang Gao, Ding-Zhu Du, Xiao-Dong Hun, and Xiaohua Jia. Rivest-vuillemin conjecture is true for monotone boolean functions with twelve variables. Discrete Mathematics, 253(1):19 - 34, 2002. Combinatorics and Algorithms. Sui-Xiang Gao, Weili Wu, Ding-Zhu Du, and Xiao-Dong Hu. The rivest-vuillemin conjecture on monotone boolean functions is true for ten variables. Journal of Complexity, 15(4):526 - 536, 1999 Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. V. King. A lower bound for the recognition of digraph properties. Combinatorica, 10(1):53-59, Mar 1990. D.J Kleitman and D.J Kwiatkowski. Further results on the aanderaarosenberg conjecture. Journal of Combinatorial Theory, Series B, 28(1):85 - 95, 1980 Jeff Kahn, Michael Saks, and Dean Sturtevant. A topological approach to evasiveness. Combinatorica, 4(4):297-306, Dec 1984. Torsten Korneffel and Eberhard Triesch. An asymptotic bound for the complexity of monotone graph properties. Combinatorica, 30(6):735- 743, Nov 2010 H.W. Lenstra, M.R. Best, and Peter van Emde Boas. A sharpened version of the aanderaa-rosenberg conjecture. Report 30/74, Mathematisch Centrum Amsterdam (1974), 01 1974. Frank H. Lutz. Examples of z-acyclic and contractible vertexhomogeneous simplicial complexes. Technical report, Discrete Comput. Geom, 2001. Frank H. Lutz. Some results related to the evasiveness conjecture. Journal of Combinatorial Theory, Series B, 81(1):110 { 124, 2001 Jiri Matousek. Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Springer Publishing Company, Incorporated, 2007. Carl A. Miller. Evasiveness of graph properties and topological fixedpoint theorems. Foundations and TrendsA^©R in Theoretical Computer Science, 7(4):337-415, 2013. Robert Oliver. Fixed-point sets of group actions on finite acyclic complexes. Comment. Math. Helv., 50:155-177, 1975. Ronald L. Rivest and Jean Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3(3):371 - 384, 1976. Gao Sui-Xiang, Hu Xiao-Dong, and Wu Weili. Nontrivial monotone weakly symmetric boolean functions with six variables are elusive. Theoretical Computer Science, 223(1):193 - 197, 1999 |
| dc.rights.en.fl_str_mv |
Attribution 4.0 International |
| dc.rights.uri.none.fl_str_mv |
http://creativecommons.org/licenses/by/4.0/ |
| dc.rights.accessrights.none.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.coar.none.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
| rights_invalid_str_mv |
Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.format.extent.none.fl_str_mv |
70 páginas |
| dc.format.mimetype.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Universidad de los Andes |
| dc.publisher.program.none.fl_str_mv |
Matemáticas |
| dc.publisher.faculty.none.fl_str_mv |
Facultad de Ciencias |
| dc.publisher.department.none.fl_str_mv |
Departamento de Matemáticas |
| publisher.none.fl_str_mv |
Universidad de los Andes |
| institution |
Universidad de los Andes |
| bitstream.url.fl_str_mv |
https://repositorio.uniandes.edu.co/bitstreams/220091a8-d61c-46b5-8426-c4267d60c741/download https://repositorio.uniandes.edu.co/bitstreams/2795820d-2d0c-4113-b523-8b68dd49d256/download https://repositorio.uniandes.edu.co/bitstreams/b06020af-6eb8-4d95-ab4c-c8ec22d78931/download https://repositorio.uniandes.edu.co/bitstreams/6fa9f852-2ae7-40b0-a8d4-18ce576604d9/download https://repositorio.uniandes.edu.co/bitstreams/5dd84f3b-9614-4f70-9542-6f6c176c8125/download https://repositorio.uniandes.edu.co/bitstreams/ef149f0f-86ef-48cd-925b-8077ec989611/download https://repositorio.uniandes.edu.co/bitstreams/de73f8c0-7813-442e-9a09-87251cacfbed/download https://repositorio.uniandes.edu.co/bitstreams/8b307d47-5816-4d53-bc5a-7b3ff86ca587/download |
| bitstream.checksum.fl_str_mv |
0325369752d7ea56bae4ee519fa53cca aa4b147dccb9cd1a3d9df082c968cf2e ae9e573a68e7f92501b6913cc846c39f 0175ea4a2d4caec4bbcc37e300941108 7f60b5640bb8121ac86fc6c3322f773e cf20c2d62558622964d700a45661e7c1 c1630b99ef952b69dd390ec9eae6cf5e eeae6f477eab7ed936026e7dbebe09f5 |
| bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 MD5 MD5 MD5 |
| repository.name.fl_str_mv |
Repositorio institucional Séneca |
| repository.mail.fl_str_mv |
adminrepositorio@uniandes.edu.co |
| _version_ |
1837005190356533248 |
| spelling |
Ángel Cárdenas, Jairo Andrésvirtual::24083-1Mantilla Acosta, Rafael JoséBogart, Tristram Charles2025-05-20T20:19:05Z2025-05-20T20:19:05Z2025-04-20https://hdl.handle.net/1992/76183instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/Deciding graph properties—Boolean functions invariant under graph isomorphism—is a foundational challenge in computational complexity. A central question is determining the minimum number of edge queries required to verify such properties in the worst case. A property is evasive if resolving it demands inspecting every potential edge. While most graph properties are non-evasive, the Evasiveness Conjecture posits that all monotone graph properties (those preserved under edge addition) are evasive. Though unresolved in general, this conjecture is proven for graphs with prime power vertex counts, a landmark result deeply tied to algebraic and topological methods. This work synthesizes advances in the Evasiveness Conjecture, providing a detailed exposition of its proof for prime power-sized graphs. Our analysis reveals unexpected connections to finite group theory and simplicial topology, underscoring how combinatorial problems of this nature straddle diverse mathematical domains. By unifying these perspectives, we clarify the conjecture’s current boundaries and the role of symmetry in computational hardness.Pregrado70 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de MatemáticasAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2The complexity of recognizing graph properties and the evasiveness conjectureLa complejidad de reconocer propiedades de grafos y la conjetura de evasividadTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPEvasiveness ConjectureMonotone graph propertiesDeterministic query complexityTopological methods in complexityMatemáticasAndrés Angel and Jerson Borja. Simplicial complexes and the evasiveness conjecture. Graduate Journal of Mathematics, 1(1), Jun 2016.Andres Angel and Jerson Borja. The evasiveness conjecture and graphs on 2p vertices. Journal of Graph Theory, 91, 05 2019.László Babai, Anandam Banerjee, Raghav Kulkarni, and Vipul Naik. Evasiveness and the distribution of prime numbers. CoRR, abs/1001.4829, 2010.Bruno Benedetti and Frank H. Lutz. The dunce hat in a minimal non-extendably collapsible 3-ball, 2009.Amit Chakrabarti, Subhash Khot, and Yaoyun Shi. Evasiveness of subgraph containment and related properties. In Afonso Ferreira and Horst Reichel, editors, STACS 2001, pages 110-120, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg.Sui-Xiang Gao, Ding-Zhu Du, Xiao-Dong Hun, and Xiaohua Jia. Rivest-vuillemin conjecture is true for monotone boolean functions with twelve variables. Discrete Mathematics, 253(1):19 - 34, 2002. Combinatorics and Algorithms.Sui-Xiang Gao, Weili Wu, Ding-Zhu Du, and Xiao-Dong Hu. The rivest-vuillemin conjecture on monotone boolean functions is true for ten variables. Journal of Complexity, 15(4):526 - 536, 1999Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.V. King. A lower bound for the recognition of digraph properties. Combinatorica, 10(1):53-59, Mar 1990.D.J Kleitman and D.J Kwiatkowski. Further results on the aanderaarosenberg conjecture. Journal of Combinatorial Theory, Series B, 28(1):85 - 95, 1980Jeff Kahn, Michael Saks, and Dean Sturtevant. A topological approach to evasiveness. Combinatorica, 4(4):297-306, Dec 1984.Torsten Korneffel and Eberhard Triesch. An asymptotic bound for the complexity of monotone graph properties. Combinatorica, 30(6):735- 743, Nov 2010H.W. Lenstra, M.R. Best, and Peter van Emde Boas. A sharpened version of the aanderaa-rosenberg conjecture. Report 30/74, Mathematisch Centrum Amsterdam (1974), 01 1974.Frank H. Lutz. Examples of z-acyclic and contractible vertexhomogeneous simplicial complexes. Technical report, Discrete Comput. Geom, 2001.Frank H. Lutz. Some results related to the evasiveness conjecture. Journal of Combinatorial Theory, Series B, 81(1):110 { 124, 2001Jiri Matousek. Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Springer Publishing Company, Incorporated, 2007.Carl A. Miller. Evasiveness of graph properties and topological fixedpoint theorems. Foundations and TrendsA^©R in Theoretical Computer Science, 7(4):337-415, 2013.Robert Oliver. Fixed-point sets of group actions on finite acyclic complexes. Comment. Math. Helv., 50:155-177, 1975.Ronald L. Rivest and Jean Vuillemin. On recognizing graph properties from adjacency matrices. Theoretical Computer Science, 3(3):371 - 384, 1976.Gao Sui-Xiang, Hu Xiao-Dong, and Wu Weili. Nontrivial monotone weakly symmetric boolean functions with six variables are elusive. Theoretical Computer Science, 223(1):193 - 197, 1999201124446Publication24652f80-186d-4801-b2b7-e275a736898bvirtual::24083-124652f80-186d-4801-b2b7-e275a736898bvirtual::24083-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0001436959virtual::24083-1ORIGINALThe complexity of recognizing graph properties and the evasiveness conjecture.pdfThe complexity of recognizing graph properties and the evasiveness conjecture.pdfapplication/pdf466978https://repositorio.uniandes.edu.co/bitstreams/220091a8-d61c-46b5-8426-c4267d60c741/download0325369752d7ea56bae4ee519fa53ccaMD51Autorización Documento de Tesis.pdfAutorización Documento de Tesis.pdfHIDEapplication/pdf391684https://repositorio.uniandes.edu.co/bitstreams/2795820d-2d0c-4113-b523-8b68dd49d256/downloadaa4b147dccb9cd1a3d9df082c968cf2eMD53LICENSElicense.txtlicense.txttext/plain; charset=utf-82535https://repositorio.uniandes.edu.co/bitstreams/b06020af-6eb8-4d95-ab4c-c8ec22d78931/downloadae9e573a68e7f92501b6913cc846c39fMD52CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8908https://repositorio.uniandes.edu.co/bitstreams/6fa9f852-2ae7-40b0-a8d4-18ce576604d9/download0175ea4a2d4caec4bbcc37e300941108MD54TEXTThe complexity of recognizing graph properties and the evasiveness conjecture.pdf.txtThe complexity of recognizing graph properties and the evasiveness conjecture.pdf.txtExtracted texttext/plain104446https://repositorio.uniandes.edu.co/bitstreams/5dd84f3b-9614-4f70-9542-6f6c176c8125/download7f60b5640bb8121ac86fc6c3322f773eMD55Autorización Documento de Tesis.pdf.txtAutorización Documento de Tesis.pdf.txtExtracted texttext/plain2031https://repositorio.uniandes.edu.co/bitstreams/ef149f0f-86ef-48cd-925b-8077ec989611/downloadcf20c2d62558622964d700a45661e7c1MD57THUMBNAILThe complexity of recognizing graph properties and the evasiveness conjecture.pdf.jpgThe complexity of recognizing graph properties and the evasiveness conjecture.pdf.jpgGenerated Thumbnailimage/jpeg5640https://repositorio.uniandes.edu.co/bitstreams/de73f8c0-7813-442e-9a09-87251cacfbed/downloadc1630b99ef952b69dd390ec9eae6cf5eMD56Autorización Documento de Tesis.pdf.jpgAutorización Documento de Tesis.pdf.jpgGenerated Thumbnailimage/jpeg11173https://repositorio.uniandes.edu.co/bitstreams/8b307d47-5816-4d53-bc5a-7b3ff86ca587/downloadeeae6f477eab7ed936026e7dbebe09f5MD581992/76183oai:repositorio.uniandes.edu.co:1992/761832025-05-21 04:02:40.545http://creativecommons.org/licenses/by/4.0/Attribution 4.0 Internationalopen.accesshttps://repositorio.uniandes.edu.coRepositorio institucional Sénecaadminrepositorio@uniandes.edu.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 |