Skip to main content
Springer Nature Link
Log in

Characterizations of Total Dual Integrality

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4513))

  • 2244 Accesses

Abstract

In this paper we provide new characterizing properties of TDI systems. A corollary is Sturmfels’ theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. A reformulation of these test-sets to polynomial ideals actually generalizes the existence of square-free monomials to arbitrary TDI systems, providing new relations between integer programming and Gröbner bases of toric ideals. We finally show that stable set polytopes of perfect graphs are characterized by a refined fan that is a triangulation consisting only of unimodular cones, a fact that endows the Weak Perfect Graph Theorem with a computationally advantageous geometric feature. Three ways of implementing the results are described and some experience about one of these is reported.

The first author was supported by a Fulbright grant and by NSF grants DMS-9983797 and DMS-0401047. The research of the second author was supported by the “Marie Curie Training Network” ADONET of the European Community.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bruns, W., Gubeladze, J.: Normality and Covering Properties of Affine Semigroups. Manuscript

    Google Scholar 

  2. Chandrasekaran, R., Tamir, A.: On the integrality of an extreme solution to pluperfect graph and balanced systems. Oper. Res. Let. 3, 215–218 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Conti, P., Traverso, C.: Buchberger algorithm and integer programming. In: Mattson, H.F., Rao, T.R.N., Mora, T. (eds.) AAECC 1991. LNCS, vol. 539, Springer, Heidelberg (1991)

    Google Scholar 

  4. Cook, W., Fonlupt, J., Schrijver, A.: An integer analogue of Carathéodory’s theorem. J. Combin. Theory B 40, 63–70 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cook, W., Lovász, L., Schrijver, A.: A polynomial-time test for total dual integrality in fixed dimension. Mathematical Programming Study 22, 64–69 (1984)

    MATH  Google Scholar 

  6. Cornuéjols, G.: Combinatorial optimization: packing and covering. CBMS-NSF regional conference series in applied mathematics, vol. 74. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  7. Chudnovsky, M., Cornuéjols, G., Xinming, L., Vušković, K.: Recognizing Berge graphs. Combinatorica 25(2), 143–186 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cox, D., Little, J., O’Shea, D.: Ideals, varieties and algorithms, 2nd edn. Springer, New York (1996)

    MATH  Google Scholar 

  9. Christof, T., Löbel, A.: PORTA (POlyhedron Representation Transformation Algorithm), available from http://www.zib.de/Optimization/Software/Porta/

  10. Chvátal, V.: On certain polytopes associated with graphs. Journal of Comb. Theory B 18, 138–154 (1975)

    Article  MATH  Google Scholar 

  11. Gerards, A., Sebő, A.: Total dual integrality implies local strong unimodularity. Mathematical Programming 38, 69–73 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fulkerson, D.R.: Anti-blocking polyhedra. Journal of Comb. Theory B 12, 50–71 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. Graver, J.: On the foundations of linear and integer programming I. Math. Programming 8, 207–226 (1975)

    Article  MathSciNet  Google Scholar 

  14. Grayson, D., Stillman, M.: Macaulay 2, a software system for research in algebraic geometry, available from http://www.math.uiuc.edu/Macaulay2/

  15. Hoşten, S., Thomas, R.R.: Gomory integer programs. Mathematical Programming B 96, 271–292 (2003)

    Article  MATH  Google Scholar 

  16. Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discrete Mathematics 2, 253–267 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. O’Shea, E.: Toric algebra and the weak perfect graph theorem. Ph.D. dissertation, University of Washington (2006)

    Google Scholar 

  18. Padberg, M.: Perfect zero-one matrices. Math. Programming 6, 180–196 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rambau, J.: TOPCOM (Triangulations Of Point Configurations and Oriented Matroids), available from http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM/

  20. Saito, M., Sturmfels, B., Takayama, N.: Gröbner deformations of hypergeometric differential equations. Algorithms and Computation in Mathematics, vol. 6. Springer, Berlin (2000)

    MATH  Google Scholar 

  21. Schrijver, A.: Theory of linear and integer programming. Wiley, Chichester (1986)

    MATH  Google Scholar 

  22. Schrijver, A.: Combinatorial optimization: polyhedra and efficiency. Algorithms and Combinatorics, vol. 24. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  23. Sebő, A.: Hilbert bases, Caratheodory’s theorem and combinatorial optimization. In: Kannan, R., Pulleyblank, W. (eds.) Integer Programming and Combinatorial Optimization, Mathematical Programming Society, University of Waterloo Press, Waterloo (1990)

    Google Scholar 

  24. Sebő, A.: Problem A.6, “TDI Matrices”, in Open Problems. Workshop on “The Perfect Graph Conjecture” (2002), http://www.aimath.org/pastworkshops/perfectgraph.html

  25. Sturmfels, B., Weismantel, R., Ziegler, G.M.: Gröbner bases of lattices, corner polyhedra, and integer programming. Beiträge Algebra Geom. 36, 281–298 (1995)

    MathSciNet  MATH  Google Scholar 

  26. Sturmfels, B.: Gröbner bases and convex polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  27. Sturmfels, B.: Algebraic recipes for integer programming. In: Hoşten, S., Lee, J., Thomas, R.R. (eds.) AMS Shortcourse: Trends in Optimization. Proceedings of Symposia in Applied Mathematics, vol. 61, American Mathematical Society, Providence (2004)

    Google Scholar 

  28. Sturmfels, B., Thomas, R.R.: Variations of cost functions in integer programming. Math. Programming 77, 357–387 (1997)

    MathSciNet  Google Scholar 

  29. Thomas, R.R.: Algebraic methods in integer programming. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA

    Edwin O’Shea

  2. CNRS, Laboratoire G-SCOP, 46, Avenue Félix Viallet, 38000 Grenoble 38031 Grenoble, Cedex 1, France

    András Sebő

Authors
  1. Edwin O’Shea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. András Sebő
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Matteo Fischetti David P. Williamson

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Berlin Heidelberg

About this paper

Cite this paper

O’Shea, E., Sebő, A. (2007). Characterizations of Total Dual Integrality. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-72792-7_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72791-0

  • Online ISBN: 978-3-540-72792-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics