Fences Are Futile: On Relaxations for the Linear Ordering Problem
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.
KeywordsLabel Edge Forward Edge Variable Gadget Clause Gadget Linear Order Problem
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