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.

• Note that you can approximate SOCPs with LPs in an efficient way which cannot be done for SDPs. So probably there will be no such a result in theory. In practice it might be that there are problems where it is beneficial to compute a SOCP approximation which gives you better results compared to a LP approximation (or the LP approximation of the SOCP model is considerably larger and takes longer to solve). Jun 26, 2019 at 15:40
• dx.doi.org/10.1287/moor.26.2.193.10561 for SOCP approximation and arxiv.org/abs/1111.0837 for some examples for the SDP case. Jun 26, 2019 at 15:55
• Would you mind write out what SOCP stands for? Jun 26, 2019 at 19:25
• Second Order Cone Program, see seas.ucla.edu/~vandenbe/publications/socp.pdf or link.springer.com/content/pdf/10.1007%2Fs10107-002-0339-5.pdf for a survey of its applications. Jun 26, 2019 at 19:35

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.