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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alberto Caprara and Matteo Fischetti, 1/2-Chvatal-Gomory cuts, Mathematical Programming, 74A, 1996, 221–235.
P. Erdös and H. Sachs, Regulare Graphe gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Univ. Halle-Wittenberg, Math. Nat. 12, 1963, 251–258.
M. X. Goemans and L. A. Hall, The Strongest Facets of the Acyclic Subgraph Polytope are Unknown, Proceedings of IPCO 1996, 415–429.
M. Grötschel, M. Jünger, and G. Reinelt, On the Maximum Acyclic Subgraph Polytope, Mathematical Programming, 33, 1985, 28–42.
M. Grötschel, M. Jünger, and G. Reinelt, Facets of the Linear Ordering Polytope, Mathematical Programming, 33, 1985, 43–60.
M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York, 1988.
Johan Håstad, Some Optimal Inapproximability Results, Proceedings of STOC, 1997.
Richard M. Karp, Reducibility Among Combinatorial Problems, Complexity of Computer Computations, Plenum Press, 1972.
Rudolf Mueller, On the Partial Order Polytope of a Digraph, Mathematical Programming, 73, 1996, 31–49.
Alantha Newman, Approximating the Maximum Acyclic Subgraph, M.S. Thesis, MIT, June 2000.
S. Poljak, Polyhedral and Eigenvalue Approximations of the Max-Cut Problem, Sets, Graphs and Numbers, Colloq. Math. Soc. Janos Bolyai 60, 1992, 569–581.
Andreas Schulz and Rudolf Mueller, The Interval Order Polytope of a Digraph, Proceedings of IPCO 1995, 50–64.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Newman, A., Vempala, S. (2001). Fences Are Futile: On Relaxations for the Linear Ordering Problem. In: Aardal, K., Gerards, B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2001. Lecture Notes in Computer Science, vol 2081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45535-3_26
Download citation
DOI: https://doi.org/10.1007/3-540-45535-3_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42225-9
Online ISBN: 978-3-540-45535-6
eBook Packages: Springer Book Archive