On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems

We show improved NP-hardness of approximating Ordering Constraint Satisfaction 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 taking a 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 subject only to P 6= NP. An extended abstract of this paper appeared in the 15th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’2013) [4]. ∗Supported by Swedish Research Council Grant 621-2012-4546. †Supported by ERC Advanced Grant 226203. ‡Supported by ERC Advanced Grant 226203. ACM Classification: F.2.2 AMS Classification: 68Q17,68W25