Worst-case uniform-demand Goemans–Linial gap

Determine the asymptotically largest possible ratio between the uniform sparsest-cut optimum and the Goemans–Linial SDP optimum over all graphs with uniform demands.

Background

The paper distinguishes the uniform-demand sparsest-cut problem from the more general setting with arbitrary capacities and demands. Although the ARV rounding theorem gives an upper bound of order O(sqrt(log n)) for graphs on n vertices, the corresponding worst-case lower bound for uniform demands is substantially smaller. Thus the precise asymptotic behavior of the uniform-demand Goemans–Linial integrality gap remains unresolved.

References

The worst-case gap in the uniform-demand case remains open: the best known lower bounds are much smaller, with Kane and Meka proving a bound of $\exp{\Omega(\sqrt{\log\log n})}$.

— Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs  (2609.05368 - Stamoulis, 4 Sep 2026) in Section 1, Introduction