Skip to main content
SpringerLink
Log in

Optimization Problems Categories

  • Conference paper
  • First Online:
Computer Aided Systems Theory — EUROCAST 2001 (EUROCAST 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2178))

Included in the following conference series:

Abstract

This work presents a categorical approach to cope with some questions originally studied within Computational Complexity Theory. It proceeds a research with theoretical emphasis, aiming at characterising the structural properties of optimization problems, related to the approximative issue, by means of Category Theory. In order to achieve it, two new categories are defined: the OPT category, which objects are optimization problems and the morphisms are the reductions between them, and the APX category, that has approximation problems as objects and approximation-preserving reductions as morphisms. Following the basic idea of categorical shape theory, a comparison mechanism between these two categories is defined and a hierarchical structure of approximation to each optimization problem can be modelled.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ASPERTI, A. and LONGO, G. Categories, Types and Structures: An Introduction to Category Theory for the Working Computer Scientist. Cambridge: MIT Press, 1991. 306p.

    MATH  Google Scholar 

  2. AUSIELLO, G.; CRESCENZI P.; GAMBOSI G.; KANN V.; MARCHETTISPACCAMELA A. and PROTASI M. Complexity and Approximation-Combinatorial Optimization Problems and their Approximability Properties. Springer-Verlag, 1999. 524 p.

    Google Scholar 

  3. BARR, M. and WELLS, C. Category Theory for Computing Science. New York: Prentice Hall, 1990. 432p.

    MATH  Google Scholar 

  4. BLUM, L.; CUCKER, F.; SHUB, M. and SMALE, S. Complexity and Real Computation. Springer-Verlag, 1998. 453p.

    Google Scholar 

  5. BOVET, D. P. and CRESCENZI, P. Introduction to the Theory of Complexity. Prentice Hall International Series in Computer Science. CA. R. HOARE SERIES EDITOR, 1994. 282p.

    Google Scholar 

  6. COOK, S. A. The Complexity of Theorem Proving Procedures. Proc. ACM Symposium on Theory of Computing, 1971. p. 151–158

    Google Scholar 

  7. CORDIER, J-M. and PORTER, T. Shape Theory: Categorical Approximations Methods. Ellis Horwood Ltd, 1990.

    Google Scholar 

  8. CRESCENZI, P. and PANCONESI, A. Completeness in Approximation Classes. Information and Computation 93, 1991. p. 241–262

    Article  MATH  MathSciNet  Google Scholar 

  9. CRESCENZI, P.; KANN, V. A Compendium of NP-Optimization Problems. T.R. SI/RR-95/02, Universita di Roma ”La Sapienza”, 1995. Available at http://www.nada.kth.se/theory/problemlist.html.

  10. GAREY, M. R. and JOHNSON, D. S. Computers and Intractability-A guide to the Theory of NP-Completeness. Bell Laboratories Murray Hill. New Jersey, 1979 — 340p.

    MATH  Google Scholar 

  11. ELLERMAN, D. P. Category Theory and Concrete Universals. ERKENNTNIS. v.28. No.3, 1988.

    Google Scholar 

  12. GOLDBLAT, R. TOPOI-The Categorial Analysis of Logic: Studies in Logic and the Foundations of Mathematics. New York: North Holland, v. 98, 1986. 551p.

    Google Scholar 

  13. HOCHBAUM, D. (Ed.) Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, 1997. 596p.

    Google Scholar 

  14. LEAL, L. A. S; MENEZES, P. B.; CLAUDIO, D. M. and TOSCANI, L. V. Categorias dos Problemas de Otimização. In: XIV SBES-Workshop on Formal Methods-WFM2000, 3. Oct. 04–06, João Pessoa, Paraíba. Proceedings... Brazil: SBC, 2000. p.104–109.

    Google Scholar 

  15. PAPADIMITRIOU, C. H. Computational Complexity. Addison-Wesley Publishing Company, 1994. 523p.

    Google Scholar 

  16. RATTRAY, C. The Shape of Complex Systems. EUROCAST’93, LNCS 763. Springer-Verlag, 1994.

    Google Scholar 

  17. RATTRAY, C. Identification and Recognition Through Shape in Complex Systems. EUROCAST’94, LNCS 1030. Springer-Verlag, 1995.

    Google Scholar 

  18. RATTRAY, C. Abstract Modelling Complex Systems. Systems: Theory and Practice. Advances in Computing Science. R. Albrecht Ed., 1998.

    Google Scholar 

  19. TREVISAN, L. Reductions and (Non-)Approximability. Università degli Studi di Roma “La Sapienza”, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department PUCRS, Mathematics, Porto Alegre, Brazil

    Liara Aparecida dos Santos Leal & Dalcidio Moraes Claudio

  2. Computing Institute FRGS, Porto Alegre, Brazil

    Paulo Blauth Menezess & Laira Vieira Toscani

Authors
  1. Liara Aparecida dos Santos Leal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paulo Blauth Menezess
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dalcidio Moraes Claudio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Laira Vieira Toscani
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Universidad de Las Palmas de Gran Canaria, Instituto Universitario de Ciencias y Tecnologías Cibernéticas, Campus de Tafira, 35017, Las Palmas de Gran Canaria, Spain

    Roberto Moreno-Díaz

  2. University of Linz, Research Institute for Symbolic Computation, 232, Schloss Hagenberg, Austria

    Bruno Buchberger

  3. University ofA Coruña, Facultad de Informática, LFCIA, Dpto. of Computer Science, Campus de Elviña, 15071, A Coruña, Spain

    José Luis Freire

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

dos Santos Leal, L.A., Blauth Menezess, P., Moraes Claudio, D., Toscani, L.V. (2001). Optimization Problems Categories. In: Moreno-Díaz, R., Buchberger, B., Luis Freire, J. (eds) Computer Aided Systems Theory — EUROCAST 2001. EUROCAST 2001. Lecture Notes in Computer Science, vol 2178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45654-6_23

Download citation

  • DOI: https://doi.org/10.1007/3-540-45654-6_23

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42959-3

  • Online ISBN: 978-3-540-45654-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation