Sum-Of-Squares Bounds via Boolean Function Analysis
We introduce a method for proving bounds on the SoS rank based on Boolean Function Analysis and Approximation Theory. We apply our technique to improve upon existing results, thus making progress towards answering several open questions.
We consider two questions by Laurent. First, finding what is the SoS rank of the linear representation of the set with no integral points. We prove that the SoS rank is between ceil[n/2] and ceil[~ n/2 +sqrt{n log{2n}} ~]. Second, proving the bounds on the SoS rank for the instance of the Min Knapsack problem. We show that the SoS rank is at least Omega(sqrt{n}) and at most ceil[{n+ 4 ceil[sqrt{n} ~]}/2]. Finally, we consider the question by Bienstock regarding the instance of the Set Cover problem. For this problem we prove the SoS rank lower bound of Omega(sqrt{n}).
SoS certificate
SoS rank
hypercube optimization
semidefinite programming
Theory of computation~Semidefinite programming
Theory of computation~Convex optimization
79:1-79:15
Track A: Algorithms, Complexity and Games
Supported by SNSF project PZ00P2_174117.
I would like to express my gratitude to Markus Schweighofer for fruitful discussions.
Adam
Kurpisz
Adam Kurpisz
ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland
10.4230/LIPIcs.ICALP.2019.79
S. Arora, S. Rao, and U. V. Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2):5:1-5:37, 2009.
B. Barak, S. B. Hopkins, J. A. Kelner, P. Kothari, A. Moitra, and A. 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.
B. Barak, J. A. Kelner, and D. Steurer. Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method. In STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 143-151, 2015.
B. Barak and A. Moitra. Noisy Tensor Completion via the Sum-of-Squares Hierarchy. In COLT 2016, New York, USA, June 23-26, 2016, pages 417-445, 2016.
B. Barak, P. Raghavendra, and D. Steurer. Rounding Semidefinite Programming Hierarchies via Global Correlation. In FOCS, pages 472-481, 2011.
B. Barak and D. Steurer. Proofs, beliefs, and algorithms through the lens of sum-of-squares, 2016. URL: https://www.sumofsquares.org.
https://www.sumofsquares.org
M. Hossein Bateni, M. Charikar, and V. Guruswami. MaxMin allocation via degree lower-bounded arborescences. In STOC, pages 543-552, 2009.
D. Bienstock and M. Zuckerberg. Subset Algebra Lift Operators for 0-1 Integer Programming. SIAM Journal on Optimization, 15(1):63-95, 2004.
S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, New York, NY, USA, 2004.
H. Chernof. A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations. Ann. Math. Statist., 23(4):493-507, December 1952.
K. K. H. Cheung. Computation of the Lasserre Ranks of Some Polytopes. Math. Oper. Res., 32(1):88-94, 2007.
E. Chlamtac. Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations. In FOCS, pages 691-701, 2007.
E. Chlamtac and G. Singh. Improved Approximation Guarantees through Higher Levels of SDP Hierarchies. In APPROX-RANDOM, pages 49-62, 2008.
E. Chlamtac and M. Tulsiani. Convex Relaxations and Integrality Gaps, pages 139-169. Springer US, Boston, MA, 2012.
W. Cook and S. Dash. On the matrix-cut rank of polyhedra. Math. Oper. Res., 26(1):19-30, 2001.
G. Cornuéjols and Y. Li. On the Rank of Mixed 0, 1 Polyhedra. In Proceedings of the 8th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2001, Utrecht, The Netherlands, pages 71-77, 2001.
M. Cygan, F. Grandoni, and M. Mastrolilli. How to Sell Hyperedges: The Hypermatching Assignment Problem. In SODA, pages 342-351, 2013.
W. F. de la Vega and C. Kenyon-Mathieu. Linear programming relaxations of maxcut. In SODA, pages 53-61, 2007.
M. X. Goemans and L. Tunçel. When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures? Math. Oper. Res., 26(4):796-815, 2001.
M. X. Goemans and D. P. Williamson. Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. J. Assoc. Comput. Mach., 42(6):1115-1145, 1995.
D. Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. Comput. Complexity, 10(2):139-154, 2001.
D. Grigoriev, E. A. Hirsch, and D. V. Pasechnik. Complexity of Semi-algebraic Proofs. In STACS, pages 419-430, 2002.
D. Grigoriev and N. Vorobjov. Complexity of Null-and Positivstellensatz proofs. Ann. Pure App. Logic, 113(1-3):153-160, 2001.
V. Guruswami and A. K. Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with psd objectives. In FOCS, pages 482-491, 2011.
S. B. Hopkins, T. Schramm, J. Shi, and D. Steurer. Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors. In STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 178-191, 2016.
P. Kothari, J. Steinhardt, and D. Steurer. Robust Moment Estimation and Improved Clustering via Sum of Squares. In STOC 2018, 2018.
P. K. Kothari, R. Mori, R. O'Donnell, and D. 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, Montreal, QC, Canada, June 19-23, 2017, pages 132-145, 2017.
A. Kurpisz, S. Leppänen, and M. Mastrolilli. Sum-of-Squares Hierarchy Lower Bounds for Symmetric Formulations. In Integer Programming and Combinatorial Optimization - 18th International Conference, IPCO 2016, Liège, Belgium, June 1-3, 2016, Proceedings, pages 362-374, 2016.
A. Kurpisz, S. Leppänen, and M. Mastrolilli. Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 78:1-78:14, 2016.
A. Kurpisz, S. Leppänen, and M. Mastrolilli. An unbounded Sum-of-Squares hierarchy integrality gap for a polynomially solvable problem. Math. Program., 166(1-2):1-17, 2017.
A. Kurpisz, S. Leppänen, and M. Mastrolilli. On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy. Math. Oper. Res., 42(1):135-143, 2017.
J. B. Lasserre. An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs. In Integer Programming and Combinatorial Optimization, 8th International IPCO Conference, Utrecht, The Netherlands, June 13-15, 2001, Proceedings, pages 293-303, 2001.
M. Laurent. A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming. Math. Oper. Res., 28(3):470-496, 2003.
M. Laurent. Lower Bound for the Number of Iterations in Semidefinite Hierarchies for the Cut Polytope. Math. Oper. Res., 28(4):871-883, 2003.
M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, volume 149 of IMA Vol. Math. Appl., pages 157-270. Springer, New York, 2009.
J. R. Lee, P. Raghavendra, and D. Steurer. Lower Bounds on the Size of Semidefinite Programming Relaxations. In STOC, pages 567-576, 2015.
T. Lee, A. Prakash, R. Wolf, and H. Yuen. On the Sum-of-Squares Degree of Symmetric Quadratic Functions. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 17:1-17:31, 2016.
A. Magen and M. Moharrami. Robust Algorithms for on Minor-Free Graphs Based on the Sherali-Adams Hierarchy. In APPROX-RANDOM, pages 258-271, 2009.
M. Mastrolilli. High Degree Sum of Squares Proofs, Bienstock-Zuckerberg Hierarchy and CG Cuts. In Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, pages 405-416, 2017.
Raghu Meka, Aaron Potechin, and Avi Wigderson. Sum-of-squares Lower Bounds for Planted Clique. In Proceedings of the 47th Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, pages 87-96, 2015.
Y. Nesterov. Global quadratic optimization via conic relaxation, pages 363-384. Kluwer Academic Publishers, 2000.
R. O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
P. Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, 2000.
R. Paturi. On the Degree of Polynomials that Approximate Symmetric Boolean Functions (Preliminary Version). In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, May 4-6, 1992, Victoria, British Columbia, Canada, pages 468-474, 1992.
A. Potechin and D. Steurer. Exact tensor completion with sum-of-squares. In COLT 2017, Amsterdam, The Netherlands, 7-10 July 2017, pages 1619-1673, 2017.
P. Raghavendra and N. Tan. Approximating CSPs with global cardinality constraints using SDP hierarchies. In SODA, pages 373-387, 2012.
T. Rivlin. The chebyshev polynomials. SERBIULA (sistema Librum 2.0), February 1974.
S. Sakaue, A. Takeda, S. Kim, and N. Ito. Exact Semidefinite Programming Relaxations with Truncated Moment Matrix for Binary Polynomial Optimization Problems. SIAM Journal on Optimization, 27(1):565-582, 2017.
T. Schramm and D. Steurer. Fast and robust tensor decomposition with applications to dictionary learning. In COLT 2017, Amsterdam, The Netherlands, 7-10 July 2017, pages 1760-1793, 2017.
Alexander A. Sherstov. Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions. Computational Complexity, 18(2):219-247, 2009. URL: http://dx.doi.org/10.1007/s00037-009-0274-4.
http://dx.doi.org/10.1007/s00037-009-0274-4
N. Shor. Class of global minimum bounds of polynomial functions. Cybernetics, 23(6):731-734, 1987.
J. Thapper and S. Zivny. The Power of Sherali-Adams Relaxations for General-Valued CSPs. SIAM J. Comput., 46(4):1241-1279, 2017.
R. Wolf. A note on quantum algorithms and the minimal degree of ε-error polynomials for symmetric functions. Quantum Information & Computation, 8(10):943-950, 2010.
Adam Kurpisz
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