Advertisement

Order

, Volume 30, Issue 1, pp 339–350 | Cite as

The Arboreal Jump Number of an Order

  • Adriana P. Figueiredo
  • Michel Habib
  • Sulamita Klein
  • Jayme Luiz Szwarcfiter
Article
  • 93 Downloads

Abstract

Let \({\mathcal P}\) be a partial order and \({\mathcal A}\) an arboreal extension of it (i.e. the Hasse diagram of \({\mathcal A}\) is a rooted tree with a unique minimal element). A jump of \({\mathcal A}\) is a relation contained in the Hasse diagram of \({\mathcal A}\), but not in the order \({\mathcal P}\). The arboreal jump number of \({\mathcal A}\) is the number of jumps contained in it. We study the problem of finding the arboreal extension of \({\mathcal P}\) having minimum arboreal jump number—a problem related to the well-known (linear) jump number problem. We describe several results for this problem, including NP-completeness, polynomial time solvable cases and bounds. We also discuss the concept of a minimal arboreal extension, namely an arboreal extension whose removal of one jump makes it no longer arboreal.

Keywords

Arboreal jump number Order extensions Greedy chain decompositions Jump number Partially ordered sets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    von Arnim, A., de la Higuera, C.: Computing the jump number on semi-orders is polynomial. Discrete Appl. Math. 51, 219–232 (1994)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bianco, L., Dell’olmo, P., Giordani, S.: An optimal algorithm to find the jump number of partially ordered sets. Comput. Optim. Appl. 8, 197–210 (1997)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bouchitté, V., Habib, M.: NP-completeness properties about linear extensions. Order 4, 143–154 (1987)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cogis, O., Habib, M.: Nombre de sauts et graphes serie-paralléls. RAIRO Inform. Theor. 13, 3–18 (1979)MathSciNetMATHGoogle Scholar
  5. 5.
    Colburn, C.J., Pulleyblank, W.R.: Minimizing setups in ordered sets of fixed width. Order 1, 225–229 (1985)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duffus, D., Rival, I., Winkler, P.: Minimizing setups for cycle-free ordered sets. Proc. Am. Math. Soc. 85, 509–513 (1982)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Faigle, U., Schrader, R.: A setup heuristic for interval orders. Oper. Res. Lett. 4, 185–188 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Felsner, S.: A 3/2-approximation algorithm for the jump number of interval orders. Order 6, 325–334 (1990)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Felsner, S.: Bounds for the Jump Number of Partially Ordered Sets. TU Berlin (1991)Google Scholar
  10. 10.
    Habib, M.: Comparability invariants. In: Pouzet, M., Richard, D. (eds.) Ordres - Description et Rôles, pp. 371–386. North-Holland, Amsterdam (1984)Google Scholar
  11. 11.
    Habib, M., Jegou, R.: N-free posets as generalizations of series-parallel posets. Discrete Appl. Math. 12, 279–291 (1985)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Habib, M., Möhring, R.: On some complexity properties of N-free posets with bounded decomposition diameter. Discrete Math. 63, 157–182 (1987)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kelly, D.: Invariants of finite comparability graphs. Order 3, 155–158 (1986)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mitas, J.: Tackling the jump number of interval orders. Order 8, 115–132 (1992)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pulleyblank, W.R.: On minimizing setups in precedence constraints scheduling. Manuscript (1981)Google Scholar
  16. 16.
    Rival, I.: Optimal linear extensions by interchanging chains. Proc. Am. Math. Soc. 89, 387–394 (1983)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Steiner, G.: On finding the jump number of a partial order by substitution decomposition. Order 2, 9–23 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Syslo, M.M.: Minimizing the jump number of a partially ordered set: a graph theoretic approach. Order 1, 7–19 (1984)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Syslo, M.M.: A graph theoretic approach to the jump number problem. In: Rival, I. (ed.) Graphs and Orders, pp. 195–215. D. Reidel, Dordrecht (1985)Google Scholar
  20. 20.
    Syslo, M.M.: An algorithm for solving the jump number problem. Discrete Math. 72, 337–346 (1988)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Syslo, M.M.: The jump number problem on interval orders - a 3/2 approximation algorithm. Discrete Math. 144, 119–130 (1995)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Adriana P. Figueiredo
    • 1
  • Michel Habib
    • 2
    • 3
  • Sulamita Klein
    • 4
    • 5
  • Jayme Luiz Szwarcfiter
    • 4
    • 5
    • 6
  1. 1.DMEUniversidade Federal do Estado do Rio de JaneiroRio de JaneiroBrazil
  2. 2.LIAFAUniversité Paris Diderot - Paris 7ParisFrance
  3. 3.CNRSUniversité Paris Diderot - Paris 7ParisFrance
  4. 4.IMUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  5. 5.COPPEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  6. 6.NCEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations