Mathematical Programming

, Volume 123, Issue 2, pp 371–394 | Cite as

On cardinality constrained cycle and path polytopes

  • Volker Kaibel
  • Rüdiger Stephan


Given a directed graph D = (N, A) and a sequence of positive integers \({1 \leq c_1 < c_2 < \cdots < c_m \leq |N|}\), we consider those path and cycle polytopes that are defined as the convex hulls of the incidence vectors simple paths and cycles of D of cardinality c p for some \({p \in \{1,\ldots,m\}}\), respectively. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. These inequalities can simply be transformed into inequalities that characterize the integer points of the undirected counterparts of cardinality constrained path and cycle polytopes. Beyond we investigate some further inequalities, in particular inequalities that are specific to odd/even paths and cycles.

Mathematics Subject Classification (2000)

05C38 90C57 90C27 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balas E., Oosten M.: On the cycle polytope of a directed graph. Networks 36(1), 34–46 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Balas, E., Stephan, R.: On the cycle polytope of a directed graph and its relaxations. Networks (submitted)Google Scholar
  3. 3.
    Bauer P.: A Polyhedral Approach to the Weighted Girth Problem. Shaker, Aachen (1995)Google Scholar
  4. 4.
    Bauer P., Linderoth J.T., Savelsbergh M.W.P.: A branch and cut approach to the cardinality constrained circuit problem. Math. Program., Ser. A 91, 307–348 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Borndörfer R., Grötschel M., Pfetsch M.E.: A column-generation approach to line planning in public transport. Transp. Sci. 41(1), 123–132 (2007)CrossRefGoogle Scholar
  6. 6.
    Boros E., Hammer P., Hartmann M., Shamir R.: Balancing problems in acyclic networks. Discrete Appl. Math. 49, 77–93 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coullard C., Pulleyblank W.R.: On cycle cones and polyhedra. Linear Algebra Appl. 114/115, 613–640 (1989)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dahl G., Gouveia L.: On the directed hop-constrained shortest path problem. Oper. Res. Lett. 32, 15–22 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dahl G., Realfsen B.: The cardinality-constrained shortest path problem in 2-graphs. Networks 36(1), 1–8 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Feige U., Seltser M.: On the densest k-subgraph problem. Technical report, Department of Applied Mathematics and Computer Science. The Weizmann Institute, Rehobot (1997)Google Scholar
  11. 11.
    Fischetti M.: Clique tree inequalites define facets of the asymmetric traveling salesman polytope. Discrete Appl. Math. 56, 9–18 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.), Polytopes Combinatorics and Computation (DMV-Seminars, pp. 43 74), Birkhäuser, Basel, 2000. See also
  13. 13.
    Grötschel, M.: Cardinality homogeneous set systems, cycles in matroids, and associated polytopes. In: Grötschel, M. (ed.) The sharpest cut. The impact of Manfred Padberg and his work. MPS/SIAM Ser. Optim., vol. 4, pp. 199–216 (2004)Google Scholar
  14. 14.
    Grötschel, M., Padberg, M.W.: Polyhedral theory. In: Lawler, E.L., et al. (ed.), The traveling salesman problem. A guided tour of combinatorial optimization. Chichester, pp. 251–305 (1985)Google Scholar
  15. 15.
    Hartmann M., Özlük Ö.: Facets of the p-cycle polytope. Discrete Appl. Math. 112, 147–178 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kaibel, V., Stephan, R.: On cardinality constrained cycle and path polytopes. ZIB-Report 07-25, Berlin, 2007. Available at
  17. 17.
    Kovalev M., Maurras J.-F., Vaxés Y.: On the convex hull of 3-cycles of the complete graph. Pesquisa Oper. 23, 99–109 (2003)Google Scholar
  18. 18.
    Maurras J.-F., Nguyen V.H.: On the linear description of the 3-cycle polytope. Eur. J. Oper. Res. 137(2), 310–325 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Maurras J.-F., Nguyen V.H.: On the linear description of the k-cycle polytope, \({PC_n^k}\). Int. Trans. Oper. Res. 8, 673–692 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)zbMATHGoogle Scholar
  21. 21.
    Nguyen, V.H.: A complete description for the k-path polyhedron. In: Proceedings of the Fifth International Conference on Modelling, Computation and Optimization in Information Systems and Management Science, Hermes Science Publishing, pp. 249–255 (2004)Google Scholar
  22. 22.
    Stephan, R.: Facets of the (s,t)-p-path polytope. arXiv: math.OC/0606308 (2006)Google Scholar
  23. 23.
    Schrijver A.: Combinatorial Optimization, vol. A. Springer, Berlin (2003)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Otto-von-Guericke Universität Magdeburg (FMA/IMO)MagdeburgGermany
  2. 2.Technische Universität BerlinBerlinGermany

Personalised recommendations