Framework-optimality conjecture for ordering CSP approximation

Prove that the relax–solve–round framework introduced in the paper—based on replacing each predicate by a tractable relaxation and then applying a strong IDU transformation—captures optimal approximation algorithms for both (i) completely satisfiable ordering CSPs and (ii) nearly satisfiable ordering CSPs with ε = 1/polylog(n).

Background

The authors propose a general framework that relaxes an instance to a tractable ordering CSP, solves the relaxation, and applies a randomized transformation (a strong IDU transformation) to obtain an ordering for the original instance. They show strong structural results about these transformations and provide algorithms to compute near-optimal ones within the framework.

They put forth a conjecture that this framework is not merely useful but optimal for the two central open questions identified in the introduction.