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Explicit SoS Lower Bounds from High-Dimensional Expanders

Authors Irit Dinur, Yuval Filmus , Prahladh Harsha , Madhur Tulsiani

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Author Details

Irit Dinur
  • Weizmann Institute of Science, Rehovot, Israel
Yuval Filmus
  • Technion - Israel Institute of Technology, Haifa, Israel
Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India
Madhur Tulsiani
  • Toyota Technological Institute at Chicago, IL, USA

Funding

  • Dinur, Irit: ERC-CoG grant number 772839.
  • Filmus, Yuval: European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC.
  • Harsha, Prahladh: Department of Atomic Energy, Government of India, under project no. RTI4001 and the Swarnajayanti fellowship, Department of Science and Technology, Government of India.
  • Tulsiani, Madhur: NSF grant CCF-1816372

Acknowledgements

Part of this work was done when the authors were visiting the Simons Institute of Theory of Computing, Berkeley for the 2019 summer cluster on "Error-Correcting Codes and High-Dimensional Expansion". We thank the Simons Institute for their kind hospitality.

Cite AsGet BibTex

Irit Dinur, Yuval Filmus, Prahladh Harsha, and Madhur Tulsiani. Explicit SoS Lower Bounds from High-Dimensional Expanders. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.38

Abstract

We construct an explicit and structured family of 3XOR instances which is hard for O(√{log n}) levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems are highly structured and can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author (FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

Subject Classification

ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Proof complexity
Keywords
  • High-dimensional expanders
  • sum-of-squares
  • integrality gaps

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References

  1. Peter Abramenko and Kenneth S. Brown. Buildings, Theory and applications, volume 248 of Graduate Texts in Mathematics. Springer, 2008. URL: http://dx.doi.org/10.1007/978-0-387-78835-7.
  2. Vedat Levi Alev, Fernando Granha Jeronimo, Dylan Quintana, Shashank Srivastava, and Madhur Tulsiani. List decoding of direct sum codes. In Proc. 31st Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1412-1425, 2020. URL: http://dx.doi.org/10.1137/1.9781611975994.85.
  3. Vedat Levi Alev, Fernando Granha Jeronimo, and Madhur Tulsiani. Approximating constraint satisfaction problems on high-dimensional expanders. In Proc. 60th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 180-201, 2019. URL: http://dx.doi.org/10.1109/FOCS.2019.00021.
  4. Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and applications. In Proc. 52nd ACM Symp. on Theory of Computing (STOC), pages 1198-1211, 2020. URL: http://dx.doi.org/10.1145/3357713.3384317.
  5. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. In Proc. 61st IEEE Symp. on Foundations of Comp. Science (FOCS), 2020. URL: http://arxiv.org/abs/2001.00303.
  6. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. In Proc. 51st ACM Symp. on Theory of Computing (STOC), pages 1-12, 2019. URL: http://dx.doi.org/10.1145/3313276.3316385.
  7. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding semidefinite programming hierarchies via global correlation. In Proc. 51st IEEE Symp. on Foundations of Comp. Science (FOCS), pages 472-481, 2011. https://eccc.weizmann.ac.il/eccc-reports/2011/TR11-065, URL: http://dx.doi.org/10.1109/FOCS.2011.95.
  8. Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. J. ACM, 48(2):149-169, 2001. (Preliminary version in 31st STOC, 1999). https://eccc.weizmann.ac.il/eccc-reports/1999/TR99-022, URL: http://dx.doi.org/10.1145/375827.375835.
  9. Martin R. Bridson and André Haefliger. Metric Spaces of Non-Positive Curvature, volume 319 of Grundlehren der mathematischen Wissenschaften. Springer, 1999. URL: http://dx.doi.org/10.1007/978-3-662-12494-9.
  10. Siu On Chan. Approximation resistance from pairwise-independent subgroups. J. ACM, 63(3):27:1-27:32, 2016. (Preliminary version in 45th STOC, 2013). https://eccc.weizmann.ac.il/eccc-reports/2012/TR12-110, URL: http://dx.doi.org/10.1145/2873054.
  11. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. (manuscript), 2020. URL: http://arxiv.org/abs/2011.02075.
  12. Yotam Dikstein and Irit Dinur. Agreement testing theorems on layered set systems. In Proc. 60th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 1495-1524, 2019. https://eccc.weizmann.ac.il/eccc-reports/2019/TR19-112, URL: http://dx.doi.org/10.1109/FOCS.2019.00088.
  13. Irit Dinur, Prahladh Harsha, Tali Kaufman, Inbal Livni Navon, and Amnon TaShma. List decoding with double samplers. In Proc. 30th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 2134-2153, 2019. https://eccc.weizmann.ac.il/eccc-reports/2018/TR18-198, URL: http://dx.doi.org/10.1137/1.9781611975482.129.
  14. Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In Proc. 58th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 974-985, 2017. https://eccc.weizmann.ac.il/eccc-reports/2017/TR17-089, URL: http://dx.doi.org/10.1109/FOCS.2017.94.
  15. Shai Evra, Konstantin Golubev, and Alexander Lubotzky. Mixing properties and the chromatic number of Ramanujan Complexes. Int. Math. Res. Not., 2015(22):11520-11548, 2015. URL: http://dx.doi.org/10.1093/imrn/rnv022.
  16. Shai Evra and Tali Kaufman. Bounded degree cosystolic expanders of every dimension. In Proc. 48th ACM Symp. on Theory of Computing (STOC), pages 36-48, 2016. URL: http://dx.doi.org/10.1145/2897518.2897543.
  17. Shai Evra, Tali Kaufman, and Gilles Zémor. Decodable quantum LDPC codes beyond the square root distance barrier using high dimensional expanders. In Proc. 61st IEEE Symp. on Foundations of Comp. Science (FOCS), 2020. URL: http://arxiv.org/abs/2004.07935.
  18. Noah Fleming, Pravesh Kothari, and Toniann Pitassi. Semialgebraic proofs and efficient algorithm design. Found. Trends Theor. Comput. Sci., 14(1-2):1-221, 2019. https://eccc.weizmann.ac.il/eccc-reports/2019/TR19-106, URL: http://dx.doi.org/10.1561/0400000086.
  19. Roy Gotlib and Tali Kaufman. Testing odd direct sums using high dimensional expanders. In Dimitris Achlioptas and László A. Végh, editors, Proc. 23rd International Workshop on Randomization and Computation (RANDOM), volume 145 of LIPIcs, pages 50:1-50:20. Schloss Dagstuhl, 2019. https://eccc.weizmann.ac.il/eccc-reports/2019/TR19-124, URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.50.
  20. Dima Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theoret. Comput. Sci., 259(1-2):613-622, 2001. URL: http://dx.doi.org/10.1016/S0304-3975(00)00157-2.
  21. Mikhael Gromov. Filling Riemannian manifolds. J. Differential Geom., 18(1):1-147, 1983. URL: http://dx.doi.org/10.4310/jdg/1214509283.
  22. Mikhail Gromov. Singularities, expanders and topology of maps. Part 2: from combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal., 20:416-–526, 2010. URL: http://dx.doi.org/10.1007/s00039-010-0073-8.
  23. Anna Gundert and Uli Wagner. On eigenvalues of random complexes. Israel J. Math., 216(1):545-582, 2016. URL: http://dx.doi.org/10.1007/s11856-016-1419-1.
  24. Venkatesan Guruswami and Ali Kemal Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives. In Proc. 51st IEEE Symp. on Foundations of Comp. Science (FOCS), pages 482-491, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.36.
  25. Larry Guth. Notes on Gromov’s systolic estimate. Geom. Dedicata, 123:113-129, 2006. URL: http://dx.doi.org/10.1007/s10711-006-9111-y.
  26. Lizhen Ji. Buildings and their applications in geometry and topology. In Differential geometry, volume 22 of Adv. Lect. Math. (ALM), pages 89-210. Int. Press, Somerville, MA, 2012. (Expanded version of Lizhen Ji. Buildings and their Applications in Geometry and Topology. Asian J. Math. 10 (2006), no. 1, 11-80. https://dx.doi.org/10.4310/AJM.2006.v10.n1.a5). URL: http://www.math.lsa.umich.edu/~lji/ji-survey-building.pdf.
  27. Tali Kaufman, David Kazhdan, and Alexander Lubotzky. Isoperimetric inequalities for ramanujan complexes and topological expanders. Geom. Funct. Anal., 26(1):250-287, 2016. (Preliminary version in 55th FOCS, 2014). URL: http://dx.doi.org/10.1007/s00039-016-0362-y.
  28. Tali Kaufman and Alexander Lubotzky. High dimensional expanders and property testing. In Moni Naor, editor, Proc. 5th Innovations in Theor. Comput. Sci. (ITCS), pages 501-506. ACM, 2014. URL: http://dx.doi.org/10.1145/2554797.2554842.
  29. Tali Kaufman and David Mass. Good distance lattices from high dimensional expanders. (manuscript), 2018. URL: http://arxiv.org/abs/1803.02849.
  30. Tali Kaufman and Ran J. Tessler. New cosystolic expanders from tensors imply explicit quantum LDPC codes with Ω(√nlog^k n) distance. (manuscript), 2020. URL: http://arxiv.org/abs/2008.09495.
  31. Pravesh K. Kothari, Ryuhei Mori, Ryan O'Donnell, and David Witmer. Sum of squares lower bounds for refuting any CSP. In Proc. 49th ACM Symp. on Theory of Computing (STOC), pages 132-145, 2017. URL: http://dx.doi.org/10.1145/3055399.3055485.
  32. Nathan Linial and Roy Meshulam. Homological connectivity of random 2-complexes. Combinatorica, 26(4):475-487, 2006. URL: http://dx.doi.org/10.1007/s00493-006-0027-9.
  33. Alexander Lubotzky and Roy Meshulam. A Mooore bound for simplicial complexes. Bull. Lond. Math. Soc., 39:353-358, 2007. URL: http://dx.doi.org/10.1112/blms/bdm003.
  34. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of Ramanujan complexes of type A_d̃. European J. Combin., 26(6):965-–993, 2005. URL: http://dx.doi.org/10.1016/j.ejc.2004.06.007.
  35. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Ramanujan complexes of type A_d̃. Israel J. Math., 149(1):267-299, 2005. URL: http://dx.doi.org/10.1007/BF02772543.
  36. Grant Schoenebeck. Linear level Lasserre lower bounds for certain k-CSPs. In Proc. 49th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 593-602, 2008. (Full version available at http://schoeneb.people.si.umich.edu/papers/LasserreNew.pdf). URL: http://dx.doi.org/10.1109/FOCS.2008.74.
  37. Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In Proc. 41st ACM Symp. on Theory of Computing (STOC), pages 303-312, 2009. https://eccc.weizmann.ac.il/eccc-reports/2008/TR08-104, URL: http://dx.doi.org/10.1145/1536414.1536457.
  38. Stefan Wenger. A short proof of Gromov’s filling inequality. Proc. Amer. Math. Soc., 136(8):2937-2941, 2008. URL: http://dx.doi.org/10.1090/S0002-9939-08-09203-4.
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