On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems

TL;DR

This paper improves the NP-hardness bounds for approximating certain Ordering Constraint Satisfaction Problems, showing some are approximation resistant and establishing new inapproximability thresholds for well-studied OCSPs.

Contribution

It provides the first examples of approximation-resistant OCSPs under P ≠ NP and improves hardness bounds for key OCSPs like Maximum Acyclic Subgraph.

Findings

Maximum Acyclic Subgraph is hard to approximate within 14/15+ε.

Maximum Betweenness is hard to approximate within 1/2+ε.

Maximum Non-Betweenness is approximation resistant.

Abstract

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 inapproximability of $14/15+\epsilon$ and $1/2+\epsilon$. An OCSP is said to be approximation resistant if it is hard to approximate better than taking a uniformly random ordering. We prove 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 $\neq$ \NP.