Constant-factor approximation for sparsest cut on Abelian Cayley graphs

Establish whether sparsest cut on finite Abelian Cayley graphs admits a constant-factor approximation, equivalently whether the Goemans–Linial relaxation has a uniformly bounded integrality gap on this graph class.

Background

Prior work cited by the paper conjectured that sparsest cut on Abelian Cayley graphs may admit a constant-factor approximation. The paper proves bounds depending on generator orders and gives an explicit infinite family with gap 16/15, but it does not establish a universal constant bound or determine the worst possible gap. The discussion explicitly states that the question is not resolved.

References

In , Oveis Gharan and Trevisan showed that for a $d$-regular Cayley Graph $G$ of an Abelian group we have that $\psi(G) ~\leq~ \mathcal O (\sqrt{d}) \cdotSDP_{GL (G)$, and it was conjectured that sparsest cut on Cayley graphs may admit a constant factor approximation ratio.

— Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs  (2609.05368 - Stamoulis, 4 Sep 2026) in Section 1, subsection “Abelian Cayley graphs and a summary of our results”; revisited in Section 6, “Discussion and open problems”

Can $\psi(G)/SDP_{GL(G)$ grow as a function of $\rho(S)$ on connected Abelian Cayley graphs i.e., is the bound $2\rho(S)$ ever close to the true gap, or is it inherently loose because averaging, the spectral lower bound, and kernel rounding cannot be tight simultaneously?

— Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs  (2609.05368 - Stamoulis, 4 Sep 2026) in Section 6, “Discussion and open problems,” first Question