Advertisement

Fences Are Futile: On Relaxations for the Linear Ordering Problem

  • Alantha Newman
  • Santosh Vempala
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2081)

Abstract

We study polyhedral relaxations for the linear ordering problem. The integrality gap for the standard linear programming relaxation is 2. Our main result is that the integrality gap remains 2 even when the standard relaxations are augmented with k-fence constraints for any k, and with k-Möbius ladder constraints for k up to 7; when augmented with k-Möbius ladder constraints for general k, the gap is at least 33/17 ≈ 1:94. Our proof is non-constructive—we obtain an extremal example via the probabilistic method. Finally, we show that no relaxation that is solvable in polynomial time can have an integrality gap less than 66/65 unless P=NP.

Keywords

Label Edge Forward Edge Variable Gadget Clause Gadget Linear Order Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alberto Caprara and Matteo Fischetti, 1/2-Chvatal-Gomory cuts, Mathematical Programming, 74A, 1996, 221–235.zbMATHMathSciNetGoogle Scholar
  2. 2.
    P. Erdös and H. Sachs, Regulare Graphe gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Univ. Halle-Wittenberg, Math. Nat. 12, 1963, 251–258.zbMATHGoogle Scholar
  3. 3.
    M. X. Goemans and L. A. Hall, The Strongest Facets of the Acyclic Subgraph Polytope are Unknown, Proceedings of IPCO 1996, 415–429.Google Scholar
  4. 4.
    M. Grötschel, M. Jünger, and G. Reinelt, On the Maximum Acyclic Subgraph Polytope, Mathematical Programming, 33, 1985, 28–42.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Grötschel, M. Jünger, and G. Reinelt, Facets of the Linear Ordering Polytope, Mathematical Programming, 33, 1985, 43–60.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York, 1988.zbMATHGoogle Scholar
  7. 7.
    Johan Håstad, Some Optimal Inapproximability Results, Proceedings of STOC, 1997.Google Scholar
  8. 8.
    Richard M. Karp, Reducibility Among Combinatorial Problems, Complexity of Computer Computations, Plenum Press, 1972.Google Scholar
  9. 9.
    Rudolf Mueller, On the Partial Order Polytope of a Digraph, Mathematical Programming, 73, 1996, 31–49.MathSciNetGoogle Scholar
  10. 10.
    Alantha Newman, Approximating the Maximum Acyclic Subgraph, M.S. Thesis, MIT, June 2000.Google Scholar
  11. 11.
    S. Poljak, Polyhedral and Eigenvalue Approximations of the Max-Cut Problem, Sets, Graphs and Numbers, Colloq. Math. Soc. Janos Bolyai 60, 1992, 569–581.MathSciNetGoogle Scholar
  12. 12.
    Andreas Schulz and Rudolf Mueller, The Interval Order Polytope of a Digraph, Proceedings of IPCO 1995, 50–64.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Alantha Newman
    • 1
  • Santosh Vempala
    • 2
  1. 1.Laboratory for Computer ScienceMassachusetts Institute of TechnologyCambridgeUK
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUK

Personalised recommendations