Document

# Sum of Squares Lower Bounds from Symmetry and a Good Story

## File

LIPIcs.ITCS.2019.61.pdf
• Filesize: 431 kB
• 20 pages

## Cite As

Aaron Potechin. Sum of Squares Lower Bounds from Symmetry and a Good Story. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 61:1-61:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ITCS.2019.61

## Abstract

In this paper, we develop machinery which makes it much easier to prove sum of squares lower bounds when the problem is symmetric under permutations of [1,n] and the unsatisfiability of our problem comes from integrality arguments, i.e. arguments that an expression must be an integer. Roughly speaking, to prove SOS lower bounds with our machinery it is sufficient to verify that the answer to the following three questions is yes: 1) Are there natural pseudo-expectation values for the problem? 2) Are these pseudo-expectation values rational functions of the problem parameters? 3) Are there sufficiently many values of the parameters for which these pseudo-expectation values correspond to the actual expected values over a distribution of solutions which is the uniform distribution over permutations of a single solution? We demonstrate our machinery on three problems, the knapsack problem analyzed by Grigoriev, the MOD 2 principle (which says that the complete graph K_n has no perfect matching when n is odd), and the following Turan type problem: Minimize the number of triangles in a graph G with a given edge density. For knapsack, we recover Grigoriev's lower bound exactly. For the MOD 2 principle, we tighten Grigoriev's linear degree sum of squares lower bound, making it exact. Finally, for the triangle problem, we prove a sum of squares lower bound for finding the minimum triangle density. This lower bound is completely new and gives a simple example where constant degree sum of squares methods have a constant factor error in estimating graph densities.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Proof complexity
##### Keywords
• Sum of squares hierarchy
• proof complexity
• graph theory
• lower bounds

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0
PDF Downloads

## References

1. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential Algorithms for Unique Games and Related Problems. J. ACM, 62(5):42:1-42:25, 2015.
2. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander Flows, Geometric Embeddings and Graph Partitioning. J. ACM, 56(2):5:1-5:37, April 2009.
3. Boaz Barak, Siu On Chan, and Pravesh Kothari. Sum of Squares Lower Bounds from Pairwise Independence. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 97-106, 2015.
4. Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh Kothari, Ankur Moitra, and Aaron Potechin. A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 428-437, 2016.
5. Boaz Barak, Jonathan A. Kelner, and David Steurer. Rounding Sum-of-squares Relaxations. In Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 31-40, 2014.
6. Boaz Barak, Jonathan A. Kelner, and David Steurer. Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 143-151, 2015.
7. Boaz Barak, Pravesh K. Kothari, and David Steurer. Quantum Entanglement, Sum of Squares, and the Log Rank Conjecture. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 975-988, 2017.
8. Boaz Barak and Ankur Moitra. Noisy Tensor Completion via the Sum-of-Squares Hierarchy. In Proceedings of the 29th Conference on Learning Theory, COLT 2016, New York, USA, June 23-26, 2016, pages 417-445, 2016.
9. Grigoriy Blekherman, João Gouveia, and James Pfeiffer. Sum of Squares on the hypercube. Mathematische Zeitschrift, 284(1-2):41-54, 2016.
10. Yash Deshpande and Andrea Montanari. Improved Sum-of-Squares Lower Bounds for Hidden Clique and Hidden Submatrix Problems. In COLT, volume 40 of JMLR Workshop and Conference Proceedings, pages 523-562. JMLR.org, 2015.
11. Karin Gatermann and Pablo Parrilo. Symmetry groups, semidefinite programs, and sums of squares. Journal of Pure and Applied Algebra, 192(1-3):95-128, 2004.
12. Rong Ge and Tengyu Ma. Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms. In APPROX-RANDOM, volume 40 of LIPIcs, pages 829-849. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015.
13. Michel X. Goemans and David P. Williamson. Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. J. ACM, 42(6):1115-1145, November 1995.
14. A. Goodman. On sets of acquaintances and strangers at any party. The American Mathematical Monthly, 66(9):778-783, 1959.
15. Dima Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, 2001.
16. Dima Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theor. Comput. Sci., 259(1-2):613-622, 2001.
17. Samuel B. Hopkins, Pravesh Kothari, Aaron Henry Potechin, Prasad Raghavendra, and Tselil Schramm. On the Integrality Gap of Degree-4 Sum of Squares for Planted Clique. ACM Trans. Algorithms, 14(3):28:1-28:31, June 2018.
18. Samuel B. Hopkins, Pravesh K. Kothari, Aaron Potechin, Prasad Raghavendra, Tselil Schramm, and David Steurer. The power of sum-of-squares for detecting hidden structures. CoRR, abs/1710.05017, 2017.
19. Samuel B. Hopkins, Tselil Schramm, Jonathan Shi, and David Steurer. Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors. In STOC, pages 178-191, 2016.
20. Pravesh K. Kothari, Ryuhei Mori, Ryan O'Donnell, and David Witmer. Sum of Squares Lower Bounds for Refuting Any CSP. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 132-145, 2017.
21. Adam Kurpisz, Samuli Leppänen, and Monaldo Mastrolilli. Sum-of-Squares Hierarchy Lower Bounds for Symmetric Formulations. In IPCO, volume 9682 of Lecture Notes in Computer Science, pages 362-374. Springer, 2016.
22. Jean B. Lasserre. Global Optimization with Polynomials and the Problem of Moments. SIAM J. on Optimization, 11(3):796-817, March 2000.
23. Monique Laurent. Lower Bound for the Number of Iterations in Semidefinite Hierarchies for the Cut Polytope. Math. Oper. Res., 28(4):871-883, 2003.
24. Troy Lee, Anupam Prakash, Ronald de Wolf, and Henry Yuen. On the Sum-of-squares Degree of Symmetric Quadratic Functions. In Proceedings of the 31st Conference on Computational Complexity, CCC '16, pages 17:1-17:31, 2016.
25. Tengyu Ma, Jonathan Shi, and David Steurer. Polynomial-Time Tensor Decompositions with Sum-of-Squares. In FOCS, pages 438-446. IEEE Computer Society, 2016.
26. Raghu Meka, Aaron Potechin, and Avi Wigderson. Sum-of-squares Lower Bounds for Planted Clique. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 87-96, 2015.
27. Yuri Nesterov. Squared functional systems and optimization problems. High Performance Optimization, pages 405-440, 2000.
28. Pablo Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000.
29. Aaron Potechin and David Steurer. Exact tensor completion with sum-of-squares. In Proceedings of the 30th Conference on Learning Theory, COLT 2017, Amsterdam, The Netherlands, 7-10 July 2017, pages 1619-1673, 2017.
30. Annie Raymond, James Saunderson, Mohit Singh, and Rekha R. Thomas. Symmetric sums of squares over k-subset hypercubes. Math. Program., 167(2):315-354, 2018.
31. Alexander A. Razborov. Flag algebras. J. Symb. Log., 72(4):1239-1282, 2007.
32. Grant Schoenebeck. Linear Level Lasserre Lower Bounds for Certain k-CSPs. In FOCS, pages 593-602. IEEE Computer Society, 2008.
33. N. Shor. An approach to obtaining global extremums in polynomial mathematical programming problems. Cybernetics and Systems Analysis, 23(5):695-700, 1987.
34. Madhur Tulsiani. CSP gaps and reductions in the lasserre hierarchy. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 303-312, 2009.
X

Feedback for Dagstuhl Publishing

### Thanks for your feedback!

Feedback submitted

### Could not send message

Please try again later or send an E-mail