We show improved NP-hardness of approximating Ordering Constraint Satis-faction Problems (OCSPs). For the two most well-studied OCSPs, Maximum Acyclic Subgraph and Maximum Betweenness, we prove NP-hard approximation factors of 14/15+ε and 1/2+ε. When it is hard to approximate an OCSP by a constant better than takinga uniformly-at-random ordering, then the OCSP is said to be approximation resistant. We show that the Maximum Non-Betweenness Problem is approximation resistant and that there are width-m approximation-resistant OCSPs accepting only a fraction 1/(m/2)! of assignments. These results provide the first examples of approximation-resistant OCSPs only to P != NP.
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