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.
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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
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DOI: https://doi.org/10.1007/3-540-45535-3_26
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