Document

On the Bit Complexity of Sum-of-Squares Proofs

File

LIPIcs.ICALP.2017.80.pdf
• Filesize: 0.54 MB
• 13 pages

Cite As

Prasad Raghavendra and Benjamin Weitz. On the Bit Complexity of Sum-of-Squares Proofs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 80:1-80:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.80

Abstract

It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm. In a recent paper, Ryan O'Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits low-degree proofs of non-negativity, but these proofs necessarily involve numbers with an exponential number of bits, causing the Ellipsoid algorithm to take exponential time. In this paper we obtain both positive and negative results on the bit complexity of SoS proofs. First, we propose a sufficient condition on a polynomial system that implies a bound on the coefficients in an SoS proof. We demonstrate that this sufficient condition is applicable for common use-cases of the SoS algorithm, such as Max-CSP, Balanced Separator, Max-Clique, Max-Bisection, and Unit-Vector constraints. On the negative side, O'Donnell asked whether every polynomial system containing Boolean constraints admits proofs of polynomial bit complexity. We answer this question in the negative, giving a counterexample system and non-negative polynomial which has degree two SoS proofs, but no SoS proof with small coefficients until degree sqrt(n).
Keywords
• Sum-of-Squares
• Combinatorial Optimization
• Proof Complexity

Metrics

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

References

1. William Adams and Philippe Loustaunau. An Introduction to Gröbner Bases. American Mathematical Society, 1994.
2. B. Barak, P. Raghavendra, and D. Steurer. Rounding semidefinite programming hierarchies via global correlation. In Proc. FOCS, pages 472-481. IEEE, 2011.
3. Boaz Barak, Fernando G. S. L. Brandao, Aram W. Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC'12, pages 307-326, New York, NY, USA, 2012. ACM. URL: http://dx.doi.org/10.1145/2213977.2214006.
4. Boaz Barak and David Steurer. Sum-of-squares proofs and the quest toward optimal algorithms. In Proceedings of the 2014 International Congress of Mathematicians. International Mathematical Union, 2014.
5. Gábor Braun, Jonah Brown-Cohen, Arefin Huq, Sebastian Pokutta, Prasad Raghavendra, Aurko Roy, Benjamin Weitz, and Daniel Zink. The Matching Problem Has No Small Symmetric SDP. In Proc. of the 27th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA'16), pages 1067-1078. SIAM, 2016. URL: http://dl.acm.org/citation.cfm?id=2884435.2884510.
6. Yuval Filmus and Elchanan Mossel. Harmonicity and invariance on slices of the boolean cube. In Proc. of the 31st Conf. on Computational Complexity (CCC'16), LIPIcs, pages 16:1-16:13, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.16.
7. Gerald B. Folland. How to integrate a polynomial over a sphere. The American Mathematical Monthly, 108(5):446-448, 2001. URL: http://www.jstor.org/stable/2695802.
8. D. Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. computational complexity, 10(2):139-154, 2001. URL: http://dx.doi.org/10.1007/s00037-001-8192-0.
9. Dima Grigoriev and Nicolai Vorobjov. Complexity of Null-and Positivstellensatz proofs. Annals of Pure and Applied Logic, 113(1-3):153-160, 2001.
10. Venkatesan Guruswami and Ali Kemal Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 482-491, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.36.
11. Jean B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796-817, 2001.
12. Jean Bernard Lasserre. Optimisation globale et théorie des moments. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 331(11):929-934, 2000.
13. Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, pages 157-270. Springer, 2009.
14. Troy Lee, Anupam Prakash, Ronald de Wolf, and Henry Yuen. On the sum-of-squares degree of symmetric quadratic functions. In Proc. of the 31st Conf. on Computational Complexity (CCC'16), LIPIcs, pages 17:1-17:31, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.17.
15. Yurii Nesterov. Squared functional systems and optimization problems. In High performance optimization, pages 405-440. Springer, 2000.
16. Ryan O'Donnell. SOS is not obviously automatizable, even approximately. Innovations in Theoretical Computer Science (ITCS), 2017.
17. Pablo A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000.
18. Naum Z. Shor. Class of global minimum bounds of polynomial functions. Cybernetics, 23(6):731-734, 1987.
X

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail