On the Power of Symmetric LP and SDP Relaxations

We study the computational power of general symmetric relaxations for combinatorial optimization problems, both in the linear programming (LP) and semidefinite programming (SDP) case. We show new connections to explicit LP and SDP relaxations, like those obtained from standard hierarchies. Concretely, for k <; n/4, we show that k-rounds of sum-of squares / Lasserre relaxations of size k(
_k
ⁿ
) achieve best-possible k approximation guarantees for Max CSPs among all symmetric SDP relaxations of size at most (
_k
ⁿ
). This result gives the first k lower bounds for symmetric SDPrelaxations of Max CSPs, and indicates that the sum-of-squares method provides the “right” SDP relaxation for this class of problems. Moreover, for k <; n/4, we show the existence of symmetric LP relaxations of size O(n
^2k
) for the traveling salesman problem that achieve per instance best-possible approximation (
_k
ⁿ
).