Lower bounds for semidefinite programming relaxations

Midwest theory day, IAS CSDM seminar, STOC 2015, MIT theory colloquium, Simons symposium, Simons A&G annual meeting. pdf video

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.