Pseudorandom Self-Reductions for NP-Complete Problems

Authors Reyad Abed Elrazik, Robert Robere, Assaf Schuster, Gal Yehuda

Thumbnail PDF


  • Filesize: 0.66 MB
  • 12 pages

Document Identifiers

Author Details

Reyad Abed Elrazik
  • Taub Faculty of Computer Science, Technion, Haifa, Israel
Robert Robere
  • School of Computer Science, McGill University, Montreal, Canada
Assaf Schuster
  • Taub Faculty of Computer Science, Technion, Haifa, Israel
Gal Yehuda
  • Taub Faculty of Computer Science, Technion, Haifa, Israel

Cite AsGet BibTex

Reyad Abed Elrazik, Robert Robere, Assaf Schuster, and Gal Yehuda. Pseudorandom Self-Reductions for NP-Complete Problems. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 65:1-65:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


A language L is random-self-reducible if deciding membership in L can be reduced (in polynomial time) to deciding membership in L for uniformly random instances. It is known that several "number theoretic" languages (such as computing the permanent of a matrix) admit random self-reductions. Feigenbaum and Fortnow showed that NP-complete languages are not non-adaptively random-self-reducible unless the polynomial-time hierarchy collapses, giving suggestive evidence that NP may not admit random self-reductions. Hirahara and Santhanam introduced a weakening of random self-reductions that they called pseudorandom self-reductions, in which a language L is reduced to a distribution that is computationally indistinguishable from the uniform distribution. They then showed that the Minimum Circuit Size Problem (MCSP) admits a non-adaptive pseudorandom self-reduction, and suggested that this gave further evidence that distinguished MCSP from standard NP-Complete problems. We show that, in fact, the Clique problem admits a non-adaptive pseudorandom self-reduction, assuming the planted clique conjecture. More generally we show the following. Call a property of graphs π hereditary if G ∈ π implies H ∈ π for every induced subgraph of G. We show that for any infinite hereditary property π, the problem of finding a maximum induced subgraph H ∈ π of a given graph G admits a non-adaptive pseudorandom self-reduction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • computational complexity
  • pseudorandomness
  • worst-case to average-case
  • self reductions
  • planted clique
  • hereditary graph family


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


  1. Martín Abadi, Joan Feigenbaum, and Joe Kilian. On hiding information from an oracle. Journal of Computer and System Sciences, 39(1):21-50, 1989. Google Scholar
  2. Adi Akavia, Oded Goldreich, Shafi Goldwasser, and Dana Moshkovitz. On basing one-way functions on np-hardness. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 701-710, 2006. Google Scholar
  3. Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. Information and Computation, 256:2-8, 2017. Google Scholar
  4. Eric Allender and Shuichi Hirahara. New insights on the (non-) hardness of circuit minimization and related problems. ACM Transactions on Computation Theory (ToCT), 11(4):1-27, 2019. Google Scholar
  5. Noga Alon, Michael Krivelevich, and Benny Sudakov. Finding a large hidden clique in a random graph. Random Struct. Algorithms, 13(3-4):457-466, 1998. Google Scholar
  6. László Babai and Sophie Laplante. Stronger separations for random-self-reducibility, rounds, and advice. In Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference)(Cat. No. 99CB36317), pages 98-104. IEEE, 1999. Google Scholar
  7. Boaz Barak, Samuel B. Hopkins, Jonathan A. Kelner, Pravesh K. Kothari, Ankur Moitra, and Aaron Potechin. A nearly tight sum-of-squares lower bound for the planted clique problem. SIAM J. Comput., 48(2):687-735, 2019. URL:
  8. Andrej Bogdanov and Luca Trevisan. On worst-case to average-case reductions for np problems. SIAM Journal on Computing, 36(4):1119-1159, 2006. Google Scholar
  9. Bela Bollobas and Andrew Thomason. Projections of bodies and hereditary properties of hypergraphs. Bull. London Math. Soc, 27:417-424, 1995. Google Scholar
  10. Herman Chernoff. A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations. The Annals of Mathematical Statistics, 23(4):493-507, 1952. Google Scholar
  11. Uriel Feige and Shimon Kogan. The hardness of approximating hereditary properties. Available on: http://research. microsoft. com/research/theory/feige/homepagefiles/hereditary. pdf, pages 1-12, 2005. Google Scholar
  12. Uriel Feige and Robert Krauthgamer. The probable value of the lovász-schrijver relaxations for maximum independent set. SIAM J. Comput., 32(2):345-370, 2003. URL:
  13. J Feigenbaum and L Fortnow. On the random-self-reducibility of complete sets, university of chicago technical report 90-22. Computer Science Department, 1990. Google Scholar
  14. Joan Feigenbaum, Lance Fortnow, Carsten Lund, and Daniel A Spielman. The power of adaptiveness and additional queries in random-self-reductions. In Computational Complexity Conference, pages 338-346, 1992. Google Scholar
  15. Oded Goldreich and Guy N Rothblum. Worst-case to average-case reductions for subclasses of p, 2020. Google Scholar
  16. 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
  17. Elad Hazan and Robert Krauthgamer. How hard is it to approximate the best nash equilibrium? SIAM J. Comput., 40(1):79-91, 2011. URL:
  18. Edith Hemaspaandra, Ashish V Naik, Mitsunori Ogihara, and Alan L Selman. P-selective sets and reducing search to decision vs self-reducibility. Journal of Computer and System Sciences, 53(2):194-209, 1996. Google Scholar
  19. Shuichi Hirahara. Non-black-box worst-case to average-case reductions within NP. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 247-258. IEEE Computer Society, 2018. URL:
  20. Shuichi Hirahara. Average-case hardness of np from exponential worst-case hardness assumptions. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 292-302, 2021. Google Scholar
  21. Shuichi Hirahara and Rahul Santhanam. On the average-case complexity of mcsp and its variants. In 32nd Computational Complexity Conference (CCC 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  22. Shuichi Hirahara and Osamu Watanabe. Limits of minimum circuit size problem as oracle. In 31st Conference on Computational Complexity (CCC 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  23. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. In The collected works of Wassily Hoeffding, pages 409-426. Springer, 1994. Google Scholar
  24. Mark Jerrum. Large cliques elude the metropolis process. Random Struct. Algorithms, 3(4):347-360, 1992. URL:
  25. Ari Juels and Marcus Peinado. Hiding cliques for cryptographic security. Des. Codes Cryptogr., 20(3):269-280, 2000. Google Scholar
  26. Richard M. Karp. Probabilistic analysis of some combinatorial search problems. Algorithms and Complexity: New Directions and Recent Results, 1976. Google Scholar
  27. Ludek Kucera. Expected complexity of graph partitioning problems. Discret. Appl. Math., 57(2-3):193-212, 1995. URL:
  28. John M Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is np-complete. Journal of Computer and System Sciences, 20(2):219-230, 1980. Google Scholar
  29. R. Lipton. New directions in testing. In Distributed Computing And Cryptography, 1989. Google Scholar
  30. Carsten Lund and Mihalis Yannakakis. The approximation of maximum subgraph problems. In International Colloquium on Automata, Languages, and Programming, pages 40-51. Springer, 1993. Google Scholar
  31. David W Matula. The largest clique size in a random graph. Department of Computer Science, Southern Methodist University Dallas, Texas …, 1976. Google Scholar
  32. Cody D Murray and R Ryan Williams. On the (non) np-hardness of computing circuit complexity. Theory of Computing, 13(1):1-22, 2017. Google Scholar
  33. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 681-690, 2006. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail