Nontrivial approximation for nearly satisfiable ordering CSPs with polylogarithmic error

Determine whether, for any given ordering constraint satisfaction problem (ordering CSP), there exists a polynomial-time algorithm that achieves an approximation value strictly greater than the random-ordering baseline α_random on (1 − ε)-satisfiable instances when ε = 1/polylog(n).

Background

Prior positive results for the nearly satisfiable regime are limited to bounded-arity precedence CSPs via the Minimum Feedback Arc Set approximation, which yields a 1 − O(ε log n log log n) guarantee. For general ordering CSPs, whether one can beat the random-ordering baseline when ε tends to zero polylogarithmically in n remains unresolved.

The authors identify this as a second basic open question guiding the development of their framework.