Skip to main content
SpringerLink
Log in

Facets and Rank of Integer Polyhedra

  • Chapter
Book cover Facets of Combinatorial Optimization

Abstract

We discuss several methods of determining all facets of “small” polytopes and polyhedra and give several criteria for classifying the facets into different facet types, so as to bring order into the multitude of facets as, e.g., produced by the application of the double description algorithm (DDA). Among the forms that we consider are the normal, irreducible and minimum support representations of facets. We study symmetries of vertex and edge figures under permissible permutations that leave the underlying polyhedron unchanged with the aim of reducing the numerical effort to find all facets efficiently. Then we introduce a new notion of the rank of facets and integer polyhedra. In the last section, we present old and new results on the facets of the symmetric traveling salesman polytope \(\mathcal{Q}_{T}^{n}\) with up to n=10 cities based on our computer calculations and state a conjecture that, in the notion of rank ρ(P) introduced here, asserts \(\rho(\mathcal{Q}_{T}^{n}) = n - 5\) for all n≥5. This conjecture is supported by our calculations up to n=9 and, possibly, n=10.

Dedicated to Martin Grötschel, my first and best former doctoral student and a personal friend for almost forty years now. Ad multos annos, Martin!

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 EPUB and 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
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Alevras, D.: Small min-cut polyhedra. Math. Oper. Res. 24, 35–49 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartels, H.G., Bartels, S.G.: The facets of the asymmetric 5-city traveling salesman problem. ZOR, Z. Oper.-Res. 33, 193–197 (1989)

    MathSciNet  MATH  Google Scholar 

  3. Boyd, S.C., Cunningham, W.H.: Small traveling salesman polytopes. Math. Oper. Res. 16, 259–271 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burger, E.: Über homogene lineare Ungleichungssysteme. Z. Angew. Math. Mech. 36, 135–139 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  5. Christof, T.: Low-dimensional 0/1 polytopes and branch-and-cut in combinatorial optimization. Ph.D. thesis, University of Heidelberg, Heidelberg, Germany (1997)

    Google Scholar 

  6. Christof, T., Reinelt, G.: Combinatorial optimization and small polytopes. Top 4, 1–64 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Christof, T., Reinelt, G.: Algorithmic aspects of using small instance relaxations in parallel branch-and-cut. Algorithmica 30, 597–629 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Christof, T., Reinelt, G.: Decomposition and parallelization techniques for enumerating the facets of 0/1 polytopes. Int. J. Comput. Geom. Appl. 11, 423–437 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Christof, T., Jünger, M., Reinelt, G.: A complete description of the traveling salesman polytope on 8 nodes. Oper. Res. Lett. 10, 497–500 (2001)

    Article  Google Scholar 

  10. Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4, 305–337 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dantzig, G.B., Fulkerson, D.R., Johnson, S.M.: Solution of a large-scale travelling salesman problem. Oper. Res. 2, 393–410 (1954)

    Article  MathSciNet  Google Scholar 

  12. Deza, M., Grötschel, M., Laurent, M.: Complete descriptions of small multicut polytopes. In: Applied Geometry and Discrete Mathematics, pp. 221–252. Am. Math. Soc., Providence (1991)

    Google Scholar 

  13. Edmonds, J.: Maximum matching and a polyhedron with 0–1 vertices. J. Res. Natl. Bur. Stand. B, Math. Math. Phys. 69, 67–92 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Euler, R., Verge, H.L.: A complete and irredundant linear description of the asymmetric traveling salesman polytope on 6 nodes. Discrete Appl. Math. 62, 193–208 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fleischmann, B.: Duale und primale Schnitthyperebenenverfahren in der ganzzahligen linearen Optimierung. Ph.D. thesis, University of Hamburg, Hamburg, Germany (1970)

    Google Scholar 

  16. Gomory, R.: An algorithm for integer solutions to linear programs. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 269–302. McGraw-Hill, New York (1963)

    Google Scholar 

  17. Grötschel, M., Padberg, M.W.: Zur Oberflächenstruktur des Travelling Salesman Polytopen. In: Zimmermann, H.J., Schub, A., Späth, H., Stoer, J. (eds.) Proc. in Operations Research, vol. 4, pp. 207–211. Physica-Verlag, Würzburg (1974)

    Google Scholar 

  18. Grötschel, M., Padberg, M.W.: On the symmetric traveling salesman problem. Technical report 7536, OR—Inst. f. Ökonometrie und Operations Research, Bonn, Germany (1975)

    Google Scholar 

  19. Grötschel, M., Padberg, M.W.: On the symmetric traveling salesman problem: parts I and II. Math. Program. 16, 265–302 (1979)

    Article  MATH  Google Scholar 

  20. Grötschel, M., Padberg, M.W.: Polyhedral theory. In: Lawler, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G., Shmoys, D.B. (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester (1985)

    Google Scholar 

  21. Grötschel, M., Jünger, M., Reinelt, G.: Facets of the linear ordering polytope. Math. Program. 33, 43–60 (1985)

    Article  MATH  Google Scholar 

  22. Grünbaum, B.: Convex Polytopes. Wiley, New York (1967)

    MATH  Google Scholar 

  23. Heller, I.: On the problem of shortest paths between points. Bull. Am. Math. Soc. 59, 551–552 (1953)

    Google Scholar 

  24. Jünger, M., Reinelt, G., Rinaldi, G.: The traveling salesman problem. In: Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds.) Network Models. Handbooks in Operations Research and Management Science, vol. 7, pp. 225–330. North-Holland, Amsterdam (1995)

    Chapter  Google Scholar 

  25. Kaibel, V.: Polyhedral combinatorics of the quadratic assignment problem. Ph.D. thesis, University of Cologne, Cologne, Germany (1997)

    Google Scholar 

  26. Kuhn, H.: On certain convex polyhedra. Bull. Am. Math. Soc. 61, 557–558 (1955)

    Google Scholar 

  27. Kuhn, H.: On asymmetric traveling salesman polytopes. Presented at the 14th Intern. Symp. on Mathematical Programming (1991)

    Google Scholar 

  28. Naddef, D., Rinaldi, G.: The crown inequalities for the symmetric traveling salesman polytope. Math. Oper. Res. 17, 308–326 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Naddef, D., Rinaldi, G.: The graphical relaxation: a new framework for the symmetric traveling salesman polytope. Math. Program. 58, 53–88 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)

    MATH  Google Scholar 

  31. Norman, R.: On the convex polyhedra of the symmetric traveling salesman problem. Bull. Am. Math. Soc. 61, 559 (1955)

    Google Scholar 

  32. Padberg, M.W.: Essays in integer programming. Ph.D. thesis, Carnegie-Mellon University, Pittsburgh, PA (1971)

    Google Scholar 

  33. Padberg, M.W.: On the facial structure of set packing polyhedra. Math. Program. 5, 199–215 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  34. Padberg, M.W.: Perfect zero-one matrices. Math. Program. 6, 180–196 (1973)

    Article  MathSciNet  Google Scholar 

  35. Padberg, M.W.: On the complexity of set packing polyhedra. Ann. Discrete Math. 1, 421–434 (1977)

    Article  MathSciNet  Google Scholar 

  36. Padberg, M.W.: Covering, packing and knapsack problems. Ann. Discrete Math. 4, 265–287 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  37. Padberg, M.W.: The boolean quadric polytope: some characteristics, facets, and relatives. Math. Program. 45, 139–172 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Padberg, M.W.: Linear Optimization and Extensions. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  39. Padberg, M.W.: Classical cuts for mixed-integer programming and branch-and-cut. Math. Methods Oper. Res. 53, 173–203 (2001). Reprinted in Ann. Oper. Res. 139, 321–352 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Padberg, M.W.: The rank of (mixed-) integer polyhedra. Math. Program., Ser. A 137, 593–599 (2011)

    Article  MathSciNet  Google Scholar 

  41. Padberg, M.W., Hong, S.: On the symmetric traveling salesman problem: a computational study. Math. Program. 12, 78–107 (1980)

    MathSciNet  MATH  Google Scholar 

  42. Padberg, M.W., Rao, M.: The traveling salesman problem and a class of polyhedra of diameter two. Math. Program. 7, 32–45 (1980)

    Article  MathSciNet  Google Scholar 

  43. Padberg, M.W., Rijal, M.: Location, Scheduling, Design and Integer Programming. Kluwer Academic, Boston (1996)

    Book  MATH  Google Scholar 

  44. Padberg, M.W., Sung, T.: A polynomial-time solution to Papadimitriou and Steiglitz’s “traps”. Oper. Res. Lett. 7, 117–125 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  45. Papadimitriou, C.H.: The adjacency relation on the traveling salesman polytope is NP-complete. Math. Program. 14, 312–324 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  46. Reinelt, G.: Personal communication (2011)

    Google Scholar 

  47. Rispoli, F., Cosares, S.: A bound of 4 for the diameter of the symmetric traveling salesman polytope. SIAM J. Appl. Math. 11, 373–380 (1998)

    MathSciNet  MATH  Google Scholar 

  48. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)

    MATH  Google Scholar 

  49. Sierksma, G., Tijssen, G.: Faces with large diameter on the symmetric traveling salesman polytope. Oper. Res. Lett. 12, 73–77 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ziegler, G.: Lectures on Polytopes. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

Supported in part by ONR grant N00014-96-0327 and by visits to Cologne U (1996) and IASI-CNR Rome (1997). This paper was presented in preliminary form in a plenary session at the XVIth ISMP in August 1997 at the EPFL in Lausanne, Switzerland, under the title “Facets, Rank of Integer Polyhedra and Other Topics”.

I wish to thank the referee for his constructive criticism.

Author information

Authors and Affiliations

  1. OptCom, 17 rue Vendôme, 13007, Marseille, France

    Manfred W. Padberg

Authors
  1. Manfred W. Padberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Computer Science, University of Cologne, Pohligstr. 1, Cologne, 50969, Germany

    Michael Jünger

  2. Dept. of Computer Science, University of Heidelberg, Im Neuenheimer Feld 326, Heidelberg, 69120, Germany

    Gerhard Reinelt

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Padberg, M.W. (2013). Facets and Rank of Integer Polyhedra. In: Jünger, M., Reinelt, G. (eds) Facets of Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38189-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation