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• W2950897465 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 $2^{n^c}$, for some constant $c > 0$. 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." @default.
• W2950897465 created "2019-06-27" @default.
• W2950897465 creator A5028879014 @default.
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• W2950897465 date "2014-11-24" @default.
• W2950897465 modified "2023-09-23" @default.
• W2950897465 title "Lower bounds on the size of semidefinite programming relaxations" @default.
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• W2950897465 hasPublicationYear "2014" @default.
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