# Lower Bounds on the Size of Semidefinite Programming Relaxations

```@article{Lee2015LowerBO,
title={Lower Bounds on the Size of Semidefinite Programming Relaxations},
author={James R. Lee and Prasad Raghavendra and David Steurer},
journal={Proceedings of the forty-seventh annual ACM symposium on Theory of Computing},
year={2015}
}```
• Published 23 November 2014
• Mathematics, Computer Science
• Proceedings of the forty-seventh annual ACM symposium on Theory of Computing
We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes…
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