On Pseudodeterministic Approximation Algorithms

Authors Peter Dixon, A. Pavan, N. V. Vinodchandran

Thumbnail PDF


  • Filesize: 424 kB
  • 11 pages

Document Identifiers

Author Details

Peter Dixon
  • Iowa State University, Ames, USA
A. Pavan
  • Iowa State University, Ames, USA
N. V. Vinodchandran
  • University of Nebraska, Lincoln, USA

Cite AsGet BibTex

Peter Dixon, A. Pavan, and N. V. Vinodchandran. On Pseudodeterministic Approximation Algorithms. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 61:1-61:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We investigate the notion of pseudodeterminstic approximation algorithms. A randomized approximation algorithm A for a function f is pseudodeterministic if for every input x there is a unique value v so that A(x) outputs v with high probability, and v is a good approximation of f(x). We show that designing a pseudodeterministic version of Stockmeyer's well known approximation algorithm for the NP-membership counting problem will yield a new circuit lower bound: if such an approximation algorithm exists, then for every k, there is a language in the complexity class ZPP^{NP}_{tt} that does not have n^k-size circuits. While we do not know how to design such an algorithm for the NP-membership counting problem, we show a general result that any randomized approximation algorithm for a counting problem can be transformed to an approximation algorithm that has a constant number of influential random bits. That is, for most settings of these influential bits, the approximation algorithm will be pseudodeterministic.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Circuit complexity
  • Approximation Algorithms
  • Circuit lower bounds
  • Pseudodeterminism


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


  1. S. Aaronson. Oracles are subtle but not malicious. In IEEE Conference on Computational Complexity, pages 340-354, 2006. Google Scholar
  2. S. Arora and B. Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. Google Scholar
  3. M. Bellare, O. Goldreich, and E. Petrank. Uniform generation of np-witnesses using an np-oracle. Inf. Comput., 163(2):510-526, 2000. Google Scholar
  4. N. H. Bshouty, R. Cleve, R. Gavaldà, S. Kannan, and C. Tamon. Oracles and queries that are sufficient for exact learning. J. Comput. Syst. Sci., 52(3):421-433, 1996. Google Scholar
  5. J-Y. Cai. S^p_2 subseteq zpp^np. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 620-629, 2001. Google Scholar
  6. J. Y. Cai, R. Lipton, L. Longpré, M. Ogihara, K. Regan, and D. Sivakumar. Communication complexity of key agreement on small ranges. In STACS, pages 38-49, 1995. Google Scholar
  7. E. Gat and S. Goldwasser. Probabilistic search algorithms with unique answers and their cryptographic applications. Electronic Colloquium on Computational Complexity (ECCC), 18:136, 2011. Google Scholar
  8. O. Goldreich, S. Goldwasser, and D. Ron. On the possibilities and limitations of pseudodeterministic algorithms. In Innovations in Theoretical Computer Science, ITCS '13, Berkeley, CA, USA, January 9-12, 2013, pages 127-138, 2013. Google Scholar
  9. S. Goldwasser and O. Grossman. Bipartite perfect matching in pseudo-deterministic NC. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 87:1-87:13, 2017. Google Scholar
  10. S. Goldwasser, O. Grossman, and D. Holden. Pseudo-deterministic proofs. CoRR, abs/1706.04641, 2017. Google Scholar
  11. O. Grossman. Finding primitive roots pseudo-deterministically. Electronic Colloquium on Computational Complexity (ECCC), 22:207, 2015. Google Scholar
  12. O. Grossman and Y. Liu. Reproducibility and pseudo-determinism in log-space. Electronic Colloquium on Computational Complexity (ECCC), 25:48, 2018. Google Scholar
  13. R. Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55(1-3):40-56, 1982. Google Scholar
  14. A. Klivans and D. Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31(5):1501-1526, 2002. Google Scholar
  15. J. Köbler and O. Watanabe. New collapse consequences of NP having small circuits. SIAM J. Comput., 28(1):311-324, 1998. Google Scholar
  16. I. Oliveira and R. Santhanam. Pseudodeterministic constructions in subexponential time. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 665-677, 2017. Google Scholar
  17. M. Saks and S. Zhou. BP _hspace(s) subseteq dspace(s^3/2). J. Comput. Syst. Sci., 58(2):376-403, 1999. Google Scholar
  18. R. Santhanam. Circuit lower bounds for merlin-arthur classes. SIAM J. Comput., 39(3):1038-1061, 2009. Google Scholar
  19. L. Stockmeyer. The complexity of approximate counting (preliminary version). In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 118-126, 1983. Google Scholar
  20. L. Stockmeyer. On approximation algorithms for #p. SIAM J. Comput., 14(4):849-861, 1985. Google Scholar
  21. N. V. Vinodchandran. A note on the circuit complexity of PP. Theor. Comput. Sci., 347(1-2):415-418, 2005. Google Scholar