# 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} }

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…

## 168 Citations

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## References

SHOWING 1-10 OF 66 REFERENCES

Towards Sharp Inapproximability For Any 2-CSP

- Computer Science48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
- 2007

It is shown how to reduce the search for a good inapproximability result to a certain numeric minimization problem, and conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly.

On the Power of Symmetric LP and SDP Relaxations

- Computer Science, Mathematics2014 IEEE 29th Conference on Computational Complexity (CCC)
- 2014

For k <; n/4, it is shown that k-rounds of sum-of squares / Lasserre relaxations of size k(kn) achieve best-possible k approximation guarantees for Max CSPs among all symmetric SDP relaxation of size at most (kn).

Optimal algorithms and inapproximability results for every CSP?

- Computer Science, MathematicsSTOC
- 2008

A generic conversion from SDP integrality gaps to UGC hardness results for every CSP is shown, which achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.

Approximability and proof complexity

- Computer Science, MathematicsSODA
- 2013

This work considers bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov, and shows that this proof is automatizable using semidefinite programming (SDP), meaning that any n-variable degree-d proof can be found in time nO(d).

The matching polytope has exponential extension complexity

- Computer ScienceSTOC
- 2014

By a known reduction this also improves the lower bound on the extension complexity for the TSP polytope from 2Ω(√n) to 2 Ω(n).

Polylogarithmic inapproximability

- Computer Science, MathematicsSTOC '03
- 2003

It is shown that for every fixed ε>0, the GROUP-STEINER-TREE problem admits no efficient log2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi polynomial Las-Vegas algorithms.

On the existence of 0/1 polytopes with high semidefinite extension complexity

- MathematicsMath. Program.
- 2015

It is shown that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension $$2^{\varOmega (n)}$$2Ω(n) and an affine space.

Hypercontractivity, sum-of-squares proofs, and their applications

- Mathematics, Computer ScienceSTOC '12
- 2012

Reductions between computing the 2->4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory and the study of Khot's Unique Games Conjecture are shown.

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

- MathematicsSIAM J. Optim.
- 2015

A representation-theoretic framework is presented to study equivariant PSD lifts of a certain class of symmetric polytopes known as orbitopes which respect the symmetries of the polytope.

Approximation Limits of Linear Programs (Beyond Hierarchies)

- Mathematics, Computer Science2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

A quantitative improvement of Razborov's rectangle corruption lemma for the high error regime gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjoint ness matrix that proves that quadratic approximations for CLIQUE require linear programs of exponential size.