XP or FPT approximation for twin-width

Establish whether there exists an XP approximation algorithm, or preferably an FPT approximation algorithm, for twin-width in the general case.

Background

Approximation algorithms are known for ordered structures, and an FPT approximation algorithm is mentioned later for permutation graphs. However, the general problem of approximating twin-width remains unresolved, with even membership in XP not established.

References

It is still wide open whether there is an XP, let alone FPT, approximation algorithm in the general case.

— Twin-width one  (2501.00991 - Ahn et al., 2 Jan 2025) in Section 1, Introduction, paragraph beginning “Twin-width is an invariant…”