Nontrivial approximation for completely satisfiable ordering CSPs

Determine whether, for any ordering constraint satisfaction problem (ordering CSP) that is not polynomially tractable, there exists a polynomial-time algorithm that, on completely satisfiable instances, achieves an approximation value strictly greater than the random-ordering baseline α_random (the expected fraction of constraints satisfied by a uniformly random permutation).

Background

The paper surveys what is known for ordering CSPs: Bodirsky–Kára’s classification identifies exactly which ordering CSPs are solvable in polynomial time on satisfiable instances, and Guruswami–Håstad–Manokaran–Raghavendra–Charikar show approximation resistance for fixed constant ε in the nearly-satisfiable regime under the Unique Games Conjecture. However, this still leaves completely satisfiable instances open.

Beyond Betweenness (where a 1/2-approximation is known), essentially no general nontrivial approximation results are known for completely satisfiable instances of NP-hard ordering CSPs. The authors frame this as a basic open question motivating their framework.