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Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs

Authors Tommaso d'Orsi , Chris Jones , Jake Ruotolo , Salil Vadhan , Jiyu Zhang

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Spectra of graphs
Keywords
  • Sparsest Cut
  • Spectral Graph Theory
  • Cayley Graphs
  • Approximation Algorithms

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Abstract

Whether or not the Sparsest Cut problem admits an efficient O(1)-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture.
Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension SD_ε(G): the smallest k such that the subspace spanned by the first k Laplacian eigenvectors contains all but ε fraction of a sparsest cut.
Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups - canonical examples of graphs with poor expansion properties.
We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a (1+ε)-approximation to the uniform Sparsest Cut of a degree-d Cayley graph over an Abelian group of size n in time n^O(1) ⋅ exp{(d/ε)^O(d)}. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of O(1)-approximate sparsest cuts has an ε-net of size exp{(d/ε)^O(d)} and that the multiplicity of λ₂ is bounded by 2^O(d). The latter bound is tight and improves on a previous bound of 2^O(d²) by Lee and Makarychev.

Cite As Get BibTex

Tommaso d'Orsi, Chris Jones, Jake Ruotolo, Salil Vadhan, and Jiyu Zhang. Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.16

Author Details

Tommaso d'Orsi
  • Bocconi University, Milan, Italy
Chris Jones
  • Bocconi University, Milan, Italy
Jake Ruotolo
  • Harvard University, Cambridge, MA, USA
Salil Vadhan
  • Harvard University, Cambridge, MA, USA
Jiyu Zhang
  • Bocconi University, Milan, Italy

Funding

JR and SV are supported in part by a Simons Investigator Award to Salil Vadhan. Work began while JR and SV were visitors at Bocconi University.
  • Jones, Chris: CJ is supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement Nos. 834861 and 101019547). CJ is also a member of the Bocconi Institute for Data Science and Analytics (BIDSA).
  • Vadhan, Salil: SV was a Visiting Researcher at the Bocconi University Department of Computing Sciences, supported by Luca Trevisan’s ERC Project GA-834861.

Acknowledgements

We deeply thank Luca Trevisan for his motivation and wisdom on this problem, and for suggesting to look into Abelian Cayley graphs. CJ and JZ thank Lucas Pesenti and Robert Wang for discussions. We thank Madhu Sudan for pointing us to [Ashikhmin et al., 2001].

References

  1. Noga Alon and Yuval Roichman. Random cayley graphs and expanders. Random Structures & Algorithms, 5(2):271-284, 1994. URL: https://doi.org/10.1002/RSA.3240050203.
  2. Reid Andersen and Kevin J Lang. An algorithm for improving graph partitions. In SODA, volume 8, pages 651-660, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347154.
  3. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. Journal of the ACM (JACM), 62(5):1-25, 2015. URL: https://doi.org/10.1145/2775105.
  4. Sanjeev Arora, Subhash A Khot, Alexandra Kolla, David Steurer, Madhur Tulsiani, and Nisheeth K Vishnoi. Unique games on expanding constraint graphs are easy. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 21-28, 2008. Google Scholar
  5. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):1-37, 2009. URL: https://doi.org/10.1145/1502793.1502794.
  6. Alexei Ashikhmin, Alexander Barg, and Serge Vladut. Linear codes with exponentially many light vectors. Journal of combinatorial theory. Series A, 96(2):396-399, 2001. URL: https://doi.org/10.1006/JCTA.2001.3206.
  7. Mitali Bafna, Boaz Barak, Pravesh K Kothari, Tselil Schramm, and David Steurer. Playing unique games on certified small-set expanders. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1629-1642, 2021. URL: https://doi.org/10.1145/3406325.3451099.
  8. Mitali Bafna, Max Hopkins, Tali Kaufman, and Shachar Lovett. High dimensional expanders: Eigenstripping, pseudorandomness, and unique games. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1069-1128. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.47.
  9. Mitali Bafna and Dor Minzer. Solving unique games over globally hypercontractive graphs. arXiv preprint arXiv:2304.07284, 2023. URL: https://doi.org/10.48550/arXiv.2304.07284.
  10. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding semidefinite programming hierarchies via global correlation. In 2011 ieee 52nd annual symposium on foundations of computer science, pages 472-481. IEEE, 2011. URL: https://doi.org/10.1109/FOCS.2011.95.
  11. Brian Benson, Peter Ralli, and Prasad Tetali. Volume growth, curvature, and buser-type inequalities in graphs. International Mathematics Research Notices, 2021(22):17091-17139, 2021. Google Scholar
  12. Emmanuel Breuillard and Matthew CH Tointon. Nilprogressions and groups with moderate growth. Advances in Mathematics, 289:1008-1055, 2016. Google Scholar
  13. Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D Sivakumar. On the hardness of approximating multicut and sparsest-cut. computational complexity, 15:94-114, 2006. URL: https://doi.org/10.1007/S00037-006-0210-9.
  14. Tobias H Colding and William P Minicozzi. Harmonic functions on manifolds. Annals of mathematics, 146(3):725-747, 1997. Google Scholar
  15. David Cushing, Supanat Kamtue, Riikka Kangaslampi, Shiping Liu, and Norbert Peyerimhoff. Curvatures, graph products and ricci flatness. Journal of Graph Theory, 96(4):522-553, 2021. URL: https://doi.org/10.1002/jgt.22630.
  16. Persi Diaconis and Laurent Saloff-Coste. Moderate growth and random walk on finite groups. Geometric & Functional Analysis GAFA, 4:1-36, 1994. Google Scholar
  17. Miroslav Fiedler. Algebraic connectivity of graphs. Czechoslovak mathematical journal, 23(2):298-305, 1973. Google Scholar
  18. Kimon Fountoulakis, Meng Liu, David F Gleich, and Michael W Mahoney. Flow-based algorithms for improving clusters: A unifying framework, software, and performance. SIAM Review, 65(1):59-143, 2023. URL: https://doi.org/10.1137/20M1333055.
  19. Joel Friedman, Ram Murty, and Jean-Pierre Tillich. Spectral estimates for abelian cayley graphs. Journal of Combinatorial Theory, Series B, 96(1):111-121, 2006. URL: https://doi.org/10.1016/J.JCTB.2005.06.012.
  20. Stephen Guattery and Gary L Miller. On the quality of spectral separators. SIAM Journal on Matrix Analysis and Applications, 19(3):701-719, 1998. URL: https://doi.org/10.1137/S0895479896312262.
  21. Anupam Gupta, Robert Krauthgamer, and James R Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings., pages 534-543. IEEE, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238226.
  22. Venkatesan Guruswami and Ali Kemal Sinop. Approximating non-uniform sparsest cut via generalized spectra. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 295-305. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.22.
  23. Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao. Equiangular lines with a fixed angle. Annals of Mathematics, 194(3):729-743, 2021. Google Scholar
  24. Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao. Spherical two-distance sets and eigenvalues of signed graphs. Combinatorica, 43(2):203-232, 2023. URL: https://doi.org/10.1007/S00493-023-00002-1.
  25. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767-775, 2002. URL: https://doi.org/10.1145/509907.510017.
  26. Subhash A Khot and Nisheeth K Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into 𝓁-1. Journal of the ACM (JACM), 62(1):1-39, 2015. URL: https://doi.org/10.1145/2629614.
  27. Bo'az Klartag, Gady Kozma, Peter Ralli, and Prasad Tetali. Discrete curvature and abelian groups. Canadian Journal of Mathematics, 68(3):655-674, 2016. Google Scholar
  28. Maria Klawe. Non-existence of one-dimensional expanding graphs. In 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981), pages 109-114. IEEE, 1981. URL: https://doi.org/10.1109/SFCS.1981.23.
  29. Bruce Kleiner. A new proof of gromov’s theorem on groups of polynomial growth. Journal of the American Mathematical Society, 23(3):815-829, 2010. Google Scholar
  30. Tsz Chiu Kwok, Lap Chi Lau, Yin Tat Lee, Shayan Oveis Gharan, and Luca Trevisan. Improved cheeger’s inequality: analysis of spectral partitioning algorithms through higher order spectral gap. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 11-20, New York, NY, USA, 2013. Association for Computing Machinery. URL: https://doi.org/10.1145/2488608.2488611.
  31. Kevin J Lang, Michael W Mahoney, and Lorenzo Orecchia. Empirical evaluation of graph partitioning using spectral embeddings and flow. In International Symposium on Experimental Algorithms, pages 197-208. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02011-7_19.
  32. James R Lee and Yury Makarychev. Eigenvalue multiplicity and volume growth. arXiv preprint arXiv:0806.1745, 2008. Google Scholar
  33. James R Lee, Shayan Oveis Gharan, and Luca Trevisan. Multiway spectral partitioning and higher-order cheeger inequalities. Journal of the ACM (JACM), 61(6):1-30, 2014. URL: https://doi.org/10.1145/2665063.
  34. Tom Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM (JACM), 46(6):787-832, 1999. URL: https://doi.org/10.1145/331524.331526.
  35. Anand Louis, Prasad Raghavendra, Prasad Tetali, and Santosh Vempala. Many sparse cuts via higher eigenvalues. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1131-1140, 2012. URL: https://doi.org/10.1145/2213977.2214079.
  36. Michael W Mahoney, Lorenzo Orecchia, and Nisheeth K Vishnoi. A spectral algorithm for improving graph partitions. Technical report, Technical report. Preprint, 2009. Google Scholar
  37. Grigorii Aleksandrovich Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii, 9(4):71-80, 1973. Google Scholar
  38. Theo McKenzie, Peter Michael Reichstein Rasmussen, and Nikhil Srivastava. Support of closed walks and second eigenvalue multiplicity of graphs. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 396-407, 2021. URL: https://doi.org/10.1145/3406325.3451129.
  39. Florentin Münch. Non-negative ollivier curvature on graphs, reverse poincaré inequality, buser inequality, liouville property, harnack inequality and eigenvalue estimates. Journal de Mathématiques Pures et Appliquées, 170:231-257, 2023. Google Scholar
  40. Shayan Oveis Gharan and Luca Trevisan. ARV on abelian cayley graphs. In Theory blog, 2021. Google Scholar
  41. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 755-764, 2010. URL: https://doi.org/10.1145/1806689.1806792.
  42. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between expansion problems. In 2012 IEEE 27th Conference on Computational Complexity, pages 64-73. IEEE, 2012. URL: https://doi.org/10.1109/CCC.2012.43.
  43. Wikipedia. Stirling’s approximation. https://en.wikipedia.org/wiki/Stirling%27s_approximation. Accessed: 2024-10-24.
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