# Lower bounds on the size of semidefinite programming relaxations

with James R. Lee, Prasad Raghavendra. **STOC 2015.**

*Best Paper Award at STOC 2015*

## abstract

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 $2^{n^{\delta}}$, for some constant $\delta > 0$. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.

Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-$O(1)$ sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a $7/8$-approximation for MAX 3 SAT.

## keywords

- semidefinite programming
- lower bounds
- constraint satisfaction problems
- sum-of-squares method
- strong relaxations
- quantum information
- machine learning