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A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph

Authors Dmitriy Kunisky, Xifan Yu

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LIPIcs.CCC.2023.30.pdf
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Author Details

Dmitriy Kunisky
  • Department of Computer Science, Yale University, New Haven, CT, USA
Xifan Yu
  • Department of Computer Science, Yale University, New Haven, CT, USA

Funding

This work was partially supported by ONR Award N00014-20-1-2335 and a Simons Investigator Award from the Simons Foundation to Daniel Spielman.

Acknowledgements

We thank Afonso Bandeira, Chris Jones, and Daniel Spielman for helpful discussions, and the anonymous reviewers for their careful reading of the paper.

Cite As Get BibTex

Dmitriy Kunisky and Xifan Yu. A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 30:1-30:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CCC.2023.30

Abstract

We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number p of vertices has value at least Ω(p^{1/3}). This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is O(polylog(p)). Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to polylog(p) terms) with high probability for the Erdős-Rényi random graph on p vertices, whose clique number is with high probability O(log(p)). We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as O(p^{1/2 - ε}) for some ε > 0, and give a matrix norm calculation indicating that the pseudocalibration construction for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "√p barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from 1/2 to 1/3.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Semidefinite programming
  • Mathematics of computing → Combinatorial optimization
Keywords
  • convex optimization
  • sum of squares
  • Paley graph
  • derandomization

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References

  1. Kwangjun Ahn, Dhruv Medarametla, and Aaron Potechin. Graph matrices: Norm bounds and applications. arXiv preprint, 2016. URL: https://arxiv.org/abs/1604.03423.
  2. Noga Alon, Michael Krivelevich, and Benny Sudakov. Finding a large hidden clique in a random graph. Random Structures & Algorithms, 13(3-4):457-466, 1998. Google Scholar
  3. Christine Bachoc, Máté Matolcsi, and Imre Z Ruzsa. Squares and difference sets in finite fields. Integers, 13:A77, 2013. Google Scholar
  4. Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh K Kothari, Ankur Moitra, and Aaron Potechin. A nearly tight sum-of-squares lower bound for the planted clique problem. SIAM Journal on Computing, 48(2):687-735, 2019. Google Scholar
  5. Béla Bollobás. Random graphs. Cambridge University Press, second edition, 2001. Google Scholar
  6. I Broere, D Döman, and JN Ridley. The clique numbers and chromatic numbers of certain Paley graphs. Quaestiones Mathematicae, 11(1):91-93, 1988. Google Scholar
  7. Fan R. K. Chung, Ronald L. Graham, and Richard M. Wilson. Quasi-random graphs. Combinatorica, 9(4):345-362, 1989. Google Scholar
  8. Ernie Croot and Vsevolod F Lev. Open problems in additive combinatorics. Additive Combinatorics, 43:207-233, 2007. Google Scholar
  9. Yash Deshpande and Andrea Montanari. Improved sum-of-squares lower bounds for hidden clique and hidden submatrix problems. In 28th Annual Conference on Learning Theory (COLT 2015), pages 523-562, 2015. Google Scholar
  10. Daniel Di Benedetto, József Solymosi, and Ethan P White. On the directions determined by a Cartesian product in an affine Galois plane. Combinatorica, 41(6):755-763, 2021. Google Scholar
  11. Irit Dinur, Yuval Filmus, Prahladh Harsha, and Madhur Tulsiani. Explicit SoS lower bounds from high-dimensional expanders. arXiv preprint, 2020. URL: https://arxiv.org/abs/2009.05218.
  12. Paul Erdös and George Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935. Google Scholar
  13. RJ Evans, JR Pulham, and J Sheehan. On the number of complete subgraphs contained in certain graphs. Journal of Combinatorial Theory, Series B, 30(3):364-371, 1981. Google Scholar
  14. Uriel Feige and Robert Krauthgamer. Finding and certifying a large hidden clique in a semirandom graph. Random Structures & Algorithms, 16(2):195-208, 2000. Google Scholar
  15. Uriel Feige and Robert Krauthgamer. The probable value of the Lovász-Schrijver relaxations for maximum independent set. SIAM Journal on Computing, 32(2):345-370, 2003. Google Scholar
  16. Laura Galli and Adam N Letchford. On the Lovász theta function and some variants. Discrete Optimization, 25:159-174, 2017. Google Scholar
  17. David Gamarnik and Ilias Zadik. The landscape of the planted clique problem: Dense subgraphs and the overlap gap property. arXiv preprint, 2019. URL: https://arxiv.org/abs/1904.07174.
  18. Sidney West Graham and CJ Ringrose. Lower bounds for least quadratic non-residues. In Analytic number theory, pages 269-309. Springer, 1990. Google Scholar
  19. Dima Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1-2):613-622, 2001. Google Scholar
  20. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2. Springer Science & Business Media, 2012. Google Scholar
  21. Nebojša Gvozdenović, Monique Laurent, and Frank Vallentin. Block-diagonal semidefinite programming hierarchies for 0/1 programming. Operations Research Letters, 37(1):27-31, 2009. Google Scholar
  22. Willem H Haemers. Interlacing eigenvalues and graphs. Linear Algebra and its Applications, 226:593-616, 1995. Google Scholar
  23. Brandon Hanson and Giorgis Petridis. Refined estimates concerning sumsets contained in the roots of unity. Proceedings of the London Mathematical Society, 122(3):353-358, 2021. Google Scholar
  24. Johan Hastad. Clique is hard to approximate within n^1 - ε. In Proceedings of 37th Conference on Foundations of Computer Science, pages 627-636. IEEE, 1996. Google Scholar
  25. Alan J Hoffman. On eigenvalues and colorings of graphs. In Bernard Harris, editor, Graph Theory and its Applications. Academic Press, 1970. Google Scholar
  26. Max Hopkins and Ting-Chun Lin. Explicit lower bounds against ω(n)-rounds of sum-of-squares. arXiv preprint, 2022. URL: https://arxiv.org/abs/2204.11469.
  27. 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. arXiv preprint, 2015. URL: https://arxiv.org/abs/1507.05230.
  28. Mark Jerrum. Large cliques elude the Metropolis process. Random Structures & Algorithms, 3(4):347-359, 1992. Google Scholar
  29. Chris Jones, Aaron Potechin, Goutham Rajendran, Madhur Tulsiani, and Jeff Xu. Sum-of-squares lower bounds for sparse independent set. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 406-416. IEEE, 2022. Google Scholar
  30. Ferenc Juhász. The asymptotic behaviour of Lovász' theta function for random graphs. Combinatorica, 2(2):153-155, 1982. Google Scholar
  31. Richard M Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. Springer, 1972. Google Scholar
  32. Richard M. Karp. The probabilistic analysis of some combinatorial search algorithms. In Algorithms and complexity: New Directions and Recent Results, 1976. Google Scholar
  33. Vladimir A Kobzar and Krishnan Mody. Revisiting block-diagonal sdp relaxations for the clique number of the paley graphs. arXiv preprint, 2023. URL: https://arxiv.org/abs/2304.08615.
  34. Luděk Kučera. Expected complexity of graph partitioning problems. Discrete Applied Mathematics, 57(2-3):193-212, 1995. Google Scholar
  35. Monique Laurent. Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope. Mathematics of Operations Research, 28(4):871-883, 2003. Google Scholar
  36. Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging Applications of Algebraic Geometry, pages 157-270. Springer, 2009. Google Scholar
  37. László Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information theory, 25(1):1-7, 1979. Google Scholar
  38. Mark Magsino, Dustin G Mixon, and Hans Parshall. Linear programming bounds for cliques in Paley graphs. In Wavelets and Sparsity XVIII, volume 11138, page 111381H. International Society for Optics and Photonics, 2019. Google Scholar
  39. Raghu Meka, Aaron Potechin, and Avi Wigderson. Sum-of-squares lower bounds for planted clique. In 47th Annual ACM Symposium on Theory of Computing (STOC 2015), pages 87-96. ACM, 2015. Google Scholar
  40. Raghu Meka and Avi Wigderson. Association schemes, non-commutative polynomial concentration, and sum-of-squares lower bounds for planted clique. In Electronic Colloquium on Computational Complexity (ECCC), volume 20, page 10, 2013. Google Scholar
  41. Hugh L Montgomery. Topics in multiplicative number theory, volume 227. Springer, 1971. Google Scholar
  42. Rudi Mrazović. A random model for the Paley graph. The Quarterly Journal of Mathematics, 68(1):193-206, 2017. Google Scholar
  43. Shuo Pang. SOS lower bound for exact planted clique. In 36th Computational Complexity Conference (CCC 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2021. Google Scholar
  44. Stanislaw Radziszowski. Small Ramsey numbers. The Electronic Journal of Combinatorics, 1000:DS1-Aug, 2011. Google Scholar
  45. Prasad Raghavendra and Tselil Schramm. Tight lower bounds for planted clique in the degree-4 SOS program. arXiv preprint, 2015. URL: https://arxiv.org/abs/1507.05136.
  46. James B Shearer. Lower bounds for small diagonal Ramsey numbers. Journal of Combinatorial Theory, Series A, 42(2):302-304, 1986. Google Scholar
  47. Chi Hoi Yip. On the clique number of Paley graphs of prime power order. Finite Fields and Their Applications, 77:101930, 2022. Google Scholar
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