Document Open Access Logo

Gap Amplification for Small-Set Expansion via Random Walks

Authors Prasad Raghavendra, Tselil Schramm

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2014.381.pdf
  • Filesize: 412 kB
  • 11 pages

Document Identifiers

Author Details

Prasad Raghavendra
Tselil Schramm

Cite As Get BibTex

Prasad Raghavendra and Tselil Schramm. Gap Amplification for Small-Set Expansion via Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 381-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.381

Abstract

In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness epsilon and soundness 1/2 is at least as difficult as Small-Set Expansion with completeness epsilon and soundness f(epsilon), for any function f(epsilon) which grows faster than (epsilon)^(1/2). We achieve this amplification via random walks--the output graph corresponds to taking random walks on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition,  the size of the instances (number of vertices) produced by the reduction remains the same.

Subject Classification

Keywords
  • Gap amplification
  • Small-Set Expansion
  • random walks
  • graph products
  • Unique Games

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. In FOCS, pages 563-572, 2010. Google Scholar
  2. Sanjeev Arora, James R. Lee, and Assaf Naor. Euclidean distortion and the sparsest cut. Journal of American Mathematical Society, 21:1-21, 2008. Google Scholar
  3. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. JACM: Journal of the ACM, 45, 1998. Google Scholar
  4. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. In Proceedings of the thirty-sixth annual ACM Symposium on Theory of Computing (STOC-04), pages 222-231, New York, June 13-15 2004. ACM Press. Google Scholar
  5. Irit Dinur and David Steurer. Analytical approach to parallel repetition. CoRR, abs/1305.1979, 2013. Google Scholar
  6. Shayan Oveis Gharan and Luca Trevisan. Approximating the expansion profile and almost optimal local graph clustering. In FOCS, pages 187-196, 2012. Google Scholar
  7. Johann Hȧstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001. Google Scholar
  8. Holenstein. Parallel repetition: Simplifications and the no-signaling case. In STOC: ACM Symposium on Theory of Computing (STOC), 2007. Google Scholar
  9. Tsz Chiu Kwok and Lap Chi Lau. Personal communication, 2014. Google Scholar
  10. Frank Thomson Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787-832, 1999. Google Scholar
  11. Anand Louis, Prasad Raghavendra, and Santosh Vempala. The complexity of approximating vertex expansion. CoRR, abs/1304.3139, 2013. Google Scholar
  12. Ryan O'Donnell and David Witmer. Markov chain methods for small-set expansion. Arxiv arXiv:1204.4688, 2012. Google Scholar
  13. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In STOC, pages 755-764, 2010. Google Scholar
  14. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between expansion problems. In IEEE Conference on Computational Complexity, pages 64-73, 2012. Google Scholar
  15. Anup Rao. Parallel repetition in projection games and a concentration bound. In Richard E. Ladner and Cynthia Dwork, editors, Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC'08), pages 1-10. ACM, 2008. Google Scholar
  16. Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763-803, June 1998. Google Scholar
  17. David Steurer. On the Complexity of Unique Games and Graph Expansion. PhD thesis, Princeton University, 2010. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact