Improve the configuration-LP approximation guarantee or establish a matching hardness threshold

Determine whether the approximation guarantee of $(1/sqrt{2}-\varepsilon)$ for Nash welfare maximization under general lexicographic valuations can be improved using a relaxation stronger than the configuration LP, or establish a matching hardness threshold for this valuation class.

Background

The paper develops a $(1/\sqrt{2}-\varepsilon)$-approximation algorithm for weighted Nash welfare under general lexicographic valuations by rounding the configuration LP. It also proves that the weighted configuration LP has an integrality gap of exactly $\sqrt{2}$ for lexicographic valuations, showing that the analysis is tight for this particular relaxation.

The unresolved issue is whether stronger relaxations can surpass the configuration-LP barrier, or whether the approximation guarantee reflects an inherent computational limitation of Nash welfare maximization under lexicographic valuations.