Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract)

  • Conference paper
  • First Online:
Algorithms and Discrete Applied Mathematics (CALDAM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10156))

Included in the following conference series:

  • 816 Accesses

Abstract

Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not “symmetric”, e.g., their graphs are not vertex transitive, not even regular.

We show that in the graph of the pedigree polytope, the quotient minimum degree over number of vertices tends to 1 as the number of cities tends to infinity.

D.O. Theis—Supported by the Estonian Research Council, ETAG (Eesti Teadusagentuur), through PUT Exploratory Grant #620, and by the European Regional Development Fund through the Estonian Center of Excellence in Computer Science, EXCS.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Similar content being viewed by others

Notes

  1. 1.

    We speak of vertices of the pedigree graph and nodes of the cycles, to limit confusion.

  2. 2.

    The reason why we use this notion of “infinite cycle” is pure convenience. It does not add complexity, but it makes many of statements and proofs less cumbersome. Indeed, instead of an infinite cycle, it is ok to just use a cycle whose length M is longer than all the lengths occuring in the particular argument. So instead of “let A be an infinite cycle, and consider \(A_k\), \(A_\ell \), \(A_n\)” you have to say “let M be a large enough integer, \(A_M\) a cycle of length M, and \(A_k\), \(A_\ell \), \(A_n\) sub-cycles of \(A_M\)”. All the little arguments (e.g., Fact 8 below) have to be done in the same way.

References

  1. Aguilera, N., Katz, R., Tolomei, P.: Vertex adjacencies in the set covering polyhedron. arXiv preprint arXiv:1406.6015 (2014)

  2. Arthanari, T.S.: On pedigree polytopes and hamiltonian cycles. Discret. Math. 306, 1474–1792 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arthanari, T.S.: Study of the pedigree polytope and a sufficiency condition for nonadjacency in the tour polytope. Discret. Optim. 10(3), 224–232 (2013). http://dx.doi.org/10.1016/j.disopt.2013.07.001

    Article  MathSciNet  Google Scholar 

  4. Arthanari, T.S., Usha, M.: An alternate formulation of the symmetric traveling salesman problem and its properties. Discret. Appl. Math. 98(3), 173–190 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discret. Math. 313(1), 67–83 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grötschel, M., Padberg, M.W.: Polyhedral theory. In: Lawler, E.L., Lenstra, J.K., Kan, A., Shmoys, D.B. (eds.) The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization, chap. 8, pp. 251–306. Wiley (1985)

    Google Scholar 

  7. Kaibel, V.: Low-dimensional faces of random 0/1-polytopes. In: Bienstock, D., Nemhauser, G. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 401–415. Springer, Heidelberg (2004). doi:10.1007/978-3-540-25960-2_30

    Chapter  Google Scholar 

  8. Kaibel, V., Pashkovich, K., Theis, D.O.: Symmetry matters for sizes of extended formulations. SIAM J. Discret. Math. 26(3), 1361–1382 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kaibel, V., Remshagen, A.: On the graph-density of random 0/1-polytopes. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) APPROX/RANDOM -2003. LNCS, vol. 2764, pp. 318–328. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45198-3_27

    Google Scholar 

  10. Makkeh, A., Pourmoradnasseri, M., Theis, D.O.: On the graph of the pedigree polytope. arXiv:1611.08431 (2016)

  11. Maksimenko, A.: The common face of some 0/1-polytopes with NP-complete non-adjacency relation. J. Math. Sci. 203(6), 823–832 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Naddef, D.: Pancyclic properties of the graph of some 0–1 polyhedra. J. Comb. Theor. Ser. B 37(1), 10–26 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Naddef, D.J., Pulleyblank, W.R.: The graphical relaxation: a new framework for the symmetric traveling salesman polytope. Math. Program. Ser. A 58(1), 53–88 (1993). http://dx.doi.org/10.1007/BF01581259

    Article  MathSciNet  MATH  Google Scholar 

  14. Naddef, D.J., Pulleyblank, W.R.: Hamiltonicity in (0–1)-polyhedra. J. Comb. Theor. Ser. B 37(1), 41–52 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Oswald, M., Reinelt, G., Theis, D.O.: On the graphical relaxation of the symmetric traveling salesman polytope. Math. Program. Ser. B 110(1), 175–193 (2007). http://dx.doi.org/10.1007/s10107-006-0060-x

    Article  MathSciNet  MATH  Google Scholar 

  16. Papadimitriou, C.H.: The adjacency relation on the traveling salesman polytope is NP-complete. Math. Program. 14(1), 312–324 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pashkovich, K., Weltge, S.: Hidden vertices in extensions of polytopes. Oper. Res. Lett. 43(2), 161–164 (2015)

    Article  MathSciNet  Google Scholar 

  18. Santos, F.: A counterexample to the Hirsch conjecture. Ann. Math. 176(1), 383–412 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sarangarajan, A.: A lower bound for adjacencies on the traveling salesman polytope. SIAM J. Discret. Math. 10(3), 431–435 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)

    MATH  Google Scholar 

  21. Sierksma, G.: The skeleton of the symmetric traveling salesman polytope. Discret. Appl. Math. 43(1), 63–74 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sierksma, G., Teunter, R.H.: Partial monotonizations of hamiltonian cycle polytopes: dimensions and diameters. Discret. Appl. Math. 105(1), 173–182 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Theis, D.O.: A note on the relationship between the graphical traveling salesman polyhedron, the symmetric traveling salesman polytope, and the metric cone. Discret. Appl. Math. 158(10), 1118–1120 (2010). http://dx.doi.org/10.1016/j.dam.2010.03.003

    Article  MathSciNet  MATH  Google Scholar 

  24. Theis, D.O.: On the facial structure of symmetric and graphical traveling salesman polyhedra. Discret. Optim. 12, 10–25 (2014). http://www.sciencedirect.com/science/article/pii/S1572528613000625

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Tartu, Ülikooli 17, 51014, Tartu, Estonia

    Abdullah Makkeh, Mozhgan Pourmoradnasseri & Dirk Oliver Theis

Authors
  1. Abdullah Makkeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mozhgan Pourmoradnasseri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dirk Oliver Theis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dirk Oliver Theis .

Editor information

Editors and Affiliations

  1. University of Lethbridge, Lethbridge, Alberta, Canada

    Daya Gaur

  2. Indian Institute of Technology Madras, Chennai, India

    N.S. Narayanaswamy

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Makkeh, A., Pourmoradnasseri, M., Theis, D.O. (2017). The Graph of the Pedigree Polytope is Asymptotically Almost Complete (Extended Abstract). In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-53007-9_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53006-2

  • Online ISBN: 978-3-319-53007-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation