# Can SOCPs approximate better than LPs?

Are there any classes of NP-hard combinatorial optimization problems where Second order cone programs (SOCP) gives a better approximation than linear programs (LP)?

I am looking for results in the flavor of Goemans and Williamson's celebrated result of approximating max-cut using semidefinite programs. But I want to use SOCP instead.

Interesting question! Unfortunately, Chan et. al. (2013) https://arxiv.org/pdf/1309.0563.pdf have shown that any polynomially sized LP relaxation of max-cut has an integrality gap of $$\frac{1}{2}$$ in the worst-case. Since, as pointed out in the comments, Ben-Tal and Nemirovski have shown that SOCPs can be approximated by polynomially-sized LPs, polynomially-sized SOCP relaxations of max-cut therefore have an integrality gap of $$\frac{1}{2}$$ in the worst-case.