A Stress-Free Sum-Of-Squares Lower Bound for Coloring

Authors Pravesh K. Kothari, Peter Manohar



PDF
Thumbnail PDF

File

LIPIcs.CCC.2021.23.pdf
  • Filesize: 0.84 MB
  • 21 pages

Document Identifiers

Author Details

Pravesh K. Kothari
  • Carnegie Mellon University, Pittsburgh, PA, USA
Peter Manohar
  • Carnegie Mellon University, Pittsburgh, PA, USA

Acknowledgements

We thank Xinyu Wu for taking part in early stages of this research, and the anonymous reviewers for providing valuable feedback.

Cite AsGet BibTex

Pravesh K. Kothari and Peter Manohar. A Stress-Free Sum-Of-Squares Lower Bound for Coloring. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CCC.2021.23

Abstract

We prove that with high probability over the choice of a random graph G from the Erdős-Rényi distribution G(n, 1/2), a natural n^{O(ε² log n)}-time, degree O(ε² log n) sum-of-squares semidefinite program cannot refute the existence of a valid k-coloring of G for k = n^{1/2 + ε}. Our result implies that the refutation guarantee of the basic semidefinite program (a close variant of the Lovász theta function) cannot be appreciably improved by a natural o(log n)-degree sum-of-squares strengthening, and this is tight up to a n^{o(1)} slack in k. To the best of our knowledge, this is the first lower bound for coloring G(n, 1/2) for even a single round strengthening of the basic SDP in any SDP hierarchy. Our proof relies on a new variant of instance-preserving non-pointwise complete reduction within SoS from coloring a graph to finding large independent sets in it. Our proof is (perhaps surprisingly) short, simple and does not require complicated spectral norm bounds on random matrices with dependent entries that have been otherwise necessary in the proofs of many similar results [Boaz Barak et al., 2016; S. B. {Hopkins} et al., 2017; Dmitriy Kunisky and Afonso S. Bandeira, 2019; Mrinalkanti Ghosh et al., 2020; Mohanty et al., 2020]. Our result formally holds for a constraint system where vertices are allowed to belong to multiple color classes; we leave the extension to the formally stronger formulation of coloring, where vertices must belong to unique colors classes, as an outstanding open problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Sum-of-Squares
  • Graph Coloring
  • Independent Set
  • 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):Art. 42, 25, 2015. URL: https://doi.org/10.1145/2775105.
  2. Sanjeev Arora, Béla Bollobás, László Lovász, and Iannis Tourlakis. Proving integrality gaps without knowing the linear program. Theory Comput., 2:19-51, 2006. URL: https://doi.org/10.4086/toc.2006.v002a002.
  3. Sanjeev Arora, Eden Chlamtac, and Moses Charikar. New approximation guarantee for chromatic number. In STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 215-224. ACM, New York, 2006. URL: https://doi.org/10.1145/1132516.1132548.
  4. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2):Art. 5, 37, 2009. URL: https://doi.org/10.1145/1502793.1502794.
  5. Mitali Bafna, Boaz Barak, Pravesh Kothari, Tselil Schramm, and David Steurer. Playing unique games on certified small-set expanders. CoRR, abs/2006.09969, 2020. URL: http://arxiv.org/abs/2006.09969.
  6. Ainesh Bakshi, Ilias Diakonikolas, He Jia, Daniel M. Kane, Pravesh K. Kothari, and Santosh S. Vempala. Robustly learning mixtures of k arbitrary gaussians. CoRR, abs/2012.02119, 2020. URL: http://arxiv.org/abs/2012.02119.
  7. Ainesh Bakshi and Pravesh Kothari. List-decodable subspace recovery via sum-of-squares. CoRR, abs/2002.05139, 2020. URL: http://arxiv.org/abs/2002.05139.
  8. Ainesh Bakshi and Pravesh Kothari. Outlier-robust clustering of non-spherical mixtures, 2020. URL: http://arxiv.org/abs/2005.02970.
  9. Jess Banks, Robert Kleinberg, and Cristopher Moore. The lovász theta function for random regular graphs and community detection in the hard regime. In APPROX-RANDOM, volume 81 of LIPIcs, pages 28:1-28:22. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  10. Jess Banks, Robert Kleinberg, and Cristopher Moore. The lovász theta function for random regular graphs and community detection in the hard regime. SIAM J. Comput., 48(3):1098-1119, 2019. URL: https://doi.org/10.1137/18M1180396.
  11. Boaz Barak, Siu On Chan, and Pravesh K. Kothari. Sum of squares lower bounds from pairwise independence [extended abstract]. In STOC'15 - Proceedings of the 2015 ACM Symposium on Theory of Computing, pages 97-106. ACM, New York, 2015. Google Scholar
  12. Boaz Barak, Samuel B. Hopkins, Jonathan A. Kelner, Pravesh Kothari, Ankur Moitra, and Aaron Potechin. A nearly tight sum-of-squares lower bound for the planted clique problem. In FOCS, pages 428-437. IEEE Computer Society, 2016. Google Scholar
  13. Boaz Barak, Pravesh K. Kothari, and David Steurer. Quantum entanglement, sum of squares, and the log rank conjecture. In STOC, pages 975-988. ACM, 2017. Google Scholar
  14. Boaz Barak and David Steurer. Proofs, beliefs, and algorithms through the lens of sum-of-squares, 2016. Lecture notes in preparation, available on URL: http://sumofsquares.org.
  15. Siavosh Benabbas, Siu On Chan, Konstantinos Georgiou, and Avner Magen. Tight gaps for vertex cover in the sherali-adams SDP hierarchy. In FSTTCS, volume 13 of LIPIcs, pages 41-54. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2011. Google Scholar
  16. Siavosh Benabbas, Konstantinos Georgiou, Avner Magen, and Madhur Tulsiani. SDP gaps from pairwise independence. Theory of Computing, 8(1):269-289, 2012. Google Scholar
  17. Siavosh Benabbas and Avner Magen. Extending SDP integrality gaps to sherali-adams with applications to quadratic programming and maxcutgain. In IPCO, volume 6080 of Lecture Notes in Computer Science, pages 299-312. Springer, 2010. Google Scholar
  18. Aditya Bhaskara, Moses Charikar, Aravindan Vijayaraghavan, Venkatesan Guruswami, and Yuan Zhou. Polynomial integrality gaps for strong SDP relaxations of densest k-subgraph. In SODA, pages 388-405. SIAM, 2012. Google Scholar
  19. B. Bollobás. The chromatic number of random graphs. Combinatorica, 8(1):49-55, 1988. URL: https://doi.org/10.1007/BF02122551.
  20. Moses Charikar, Konstantin Makarychev, and Yury Makarychev. Integrality gaps for Sherali-Adams relaxations. In STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing, pages 283-292. ACM, New York, 2009. Google Scholar
  21. Eden Chlamtac. Approximation algorithms using hierarchies of semidefinite programming relaxations. In FOCS, pages 691-701. IEEE Computer Society, 2007. Google Scholar
  22. Eden Chlamtac and Madhur Tulsiani. Convex relaxations and integrality gaps. In Handbook on semidefinite, conic and polynomial optimization, volume 166 of Internat. Ser. Oper. Res. Management Sci., pages 139-169. Springer, New York, 2012. URL: https://doi.org/10.1007/978-1-4614-0769-0_6.
  23. Amin Coja-Oghlan. The lovász number of random graphs. Comb. Probab. Comput., 14(4):439-465, 2005. URL: https://doi.org/10.1017/S0963548305006826.
  24. Ilias Diakonikolas, Samuel Hopkins, Daniel Kane, and Sushrut Karmalkar. Robustly learning any clusterable mixture of gaussians. Personal Communication, 2020. Google Scholar
  25. Tommaso d'Orsi, Pravesh K. Kothari, Gleb Novikov, and David Steurer. Sparse PCA: algorithms, adversarial perturbations and certificates. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 553-564. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00058.
  26. Noah Fleming, Pravesh Kothari, and Toniann Pitassi. Semialgebraic proofs and efficient algorithm design. Foundations and Trends® in Theoretical Computer Science, 14(1-2):1-221, 2019. URL: https://doi.org/10.1561/0400000086.
  27. Mrinal Kanti Ghosh and Madhur Tulsiani. From weak to strong LP gaps for all csps. In Computational Complexity Conference, volume 79 of LIPIcs, pages 11:1-11:27. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  28. Mrinalkanti Ghosh, Fernando Granha Jeronimo, Chris Jones, Aaron Potechin, and Goutham Rajendran. Sum-of-squares lower bounds for sherrington-kirkpatrick via planted affine planes, 2020. URL: http://arxiv.org/abs/2009.01874.
  29. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach., 42(6):1115-1145, 1995. URL: https://doi.org/10.1145/227683.227684.
  30. D. Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. Comput. Complexity, 10(2):139-154, 2001. URL: https://doi.org/10.1007/s00037-001-8192-0.
  31. Dima Grigoriev and Nicolai Vorobjov. Complexity of Null- and Positivstellensatz proofs. Ann. Pure Appl. Logic, 113(1-3):153-160, 2002. First St. Petersburg Conference on Days of Logic and Computability (1999). URL: https://doi.org/10.1016/S0168-0072(01)00055-0.
  32. Max Hopkins, Tali Kaufman, and Shachar Lovett. High dimensional expanders: Random walks, pseudorandomness, and unique games. CoRR, abs/2011.04658, 2020. URL: http://arxiv.org/abs/2011.04658.
  33. S. B. Hopkins, P. K. Kothari, A. Potechin, P. Raghavendra, T. Schramm, and D. Steurer. The power of sum-of-squares for detecting hidden structures. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 720-731, 2017. URL: https://doi.org/10.1109/FOCS.2017.72.
  34. Sam B. Hopkins and Jerry Li. Mixture models, robustness, and sum of squares proofs, 2017. Google Scholar
  35. Samuel B. Hopkins, Pravesh K. Kothari, and Aaron Potechin. Sos and planted clique: Tight analysis of MPW moments at all degrees and an optimal lower bound at degree four. CoRR, abs/1507.05230, 2015. URL: http://arxiv.org/abs/1507.05230.
  36. Samuel B. Hopkins, Jonathan Shi, and David Steurer. Tensor principal component analysis via sum-of-square proofs. In COLT, volume 40 of JMLR Workshop and Conference Proceedings, pages 956-1006. JMLR.org, 2015. Google Scholar
  37. Samuel B. Hopkins and David Steurer. Efficient bayesian estimation from few samples: Community detection and related problems. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 379-390. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.42.
  38. David R. Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. J. ACM, 45(2):246-265, 1998. Google Scholar
  39. Sushrut Karmalkar, Adam R. Klivans, and Pravesh K. Kothari. List-decodable linear regression. CoRR, abs/1905.05679, 2019. URL: http://arxiv.org/abs/1905.05679.
  40. Pravesh Kothari, Ryan O'Donnell, and Tselil Schramm. SOS lower bounds with hard constraints: think global, act local. CoRR, abs/1809.01207, 2018. URL: http://arxiv.org/abs/1809.01207.
  41. Pravesh K. Kothari and Ruta Mehta. Sum-of-squares meets nash: lower bounds for finding any equilibrium. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 1241-1248. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188892.
  42. Pravesh K. Kothari, Ryuhei Mori, Ryan O'Donnell, and David Witmer. Sum of squares lower bounds for refuting any CSP. In STOC, pages 132-145. ACM, 2017. Google Scholar
  43. Pravesh K. Kothari and Jacob Steinhardt. Better agnostic clustering via relaxed tensor norms, 2017. Google Scholar
  44. Pravesh K. Kothari and David Steurer. Outlier-robust moment-estimation via sum-of-squares. CoRR, abs/1711.11581, 2017. URL: http://arxiv.org/abs/1711.11581.
  45. Dmitriy Kunisky and Afonso S. Bandeira. A tight degree 4 sum-of-squares lower bound for the sherrington-kirkpatrick hamiltonian. CoRR, abs/1907.11686, 2019. URL: http://arxiv.org/abs/1907.11686.
  46. Dmitriy Kunisky, Alexander S. Wein, and Afonso S. Bandeira. Notes on computational hardness of hypothesis testing: Predictions using the low-degree likelihood ratio. CoRR, abs/1907.11636, 2019. URL: http://arxiv.org/abs/1907.11636.
  47. Jean Bernard Lasserre. Optimisation globale et théorie des moments. C. R. Acad. Sci. Paris Sér. I Math., 331(11):929-934, 2000. URL: https://doi.org/10.1016/S0764-4442(00)01750-X.
  48. Allen Liu and Ankur Moitra. Settling the robust learnability of mixtures of gaussians. CoRR, abs/2011.03622, 2020. URL: http://arxiv.org/abs/2011.03622.
  49. László Lovász and Alexander Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization, 1(2):166-190, 1991. Google Scholar
  50. Tengyu Ma, Jonathan Shi, and David Steurer. Polynomial-time tensor decompositions with sum-of-squares. In FOCS, pages 438-446. IEEE Computer Society, 2016. Google Scholar
  51. Sidhanth Mohanty, Prasad Raghavendra, and Jeff Xu. Lifting sum-of-squares lower bounds: Degree-2 to degree-4. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, page 840–853, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384319.
  52. Pablo A Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000. Google Scholar
  53. Prasad Raghavendra and Tselil Schramm. Tight lower bounds for planted clique in the degree-4 SOS program. CoRR, abs/1507.05136, 2015. URL: http://arxiv.org/abs/1507.05136.
  54. Prasad Raghavendra and David Steurer. Integrality gaps for strong SDP relaxations of Unique Games. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2009, pages 575-585. IEEE Computer Soc., Los Alamitos, CA, 2009. URL: https://doi.org/10.1109/FOCS.2009.73.
  55. Prasad Raghavendra and Morris Yau. List decodable learning via sum of squares. CoRR, abs/1905.04660, 2019. URL: http://arxiv.org/abs/1905.04660.
  56. Prasad Raghavendra and Morris Yau. List decodable subspace recovery, 2020. URL: http://arxiv.org/abs/2002.03004.
  57. Grant Schoenebeck. Linear level lasserre lower bounds for certain k-csps. In FOCS, pages 593-602. IEEE Computer Society, 2008. Google Scholar
  58. Grant Schoenebeck, Luca Trevisan, and Madhur Tulsiani. Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut. In STOC'07 - Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 302-310. ACM, New York, 2007. URL: https://doi.org/10.1145/1250790.1250836.
  59. Hanif D. Sherali and Warren P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math., 3(3):411-430, 1990. URL: https://doi.org/10.1137/0403036.
  60. Madhur Tulsiani. CSP gaps and reductions in the lasserre hierarchy. In STOC, pages 303-312. ACM, 2009. Google Scholar
  61. Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy [extended abstract]. In STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing, pages 303-312. ACM, New York, 2009. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail