TFNP: An Update

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)

Abstract

The class TFNP was introduced a quarter of a century ago to capture problems in NP that have a witness for all inputs. A decade ago, this line of research culminated in the proof that the Nash equilibrium problem is complete for the subclass PPAD. Here we review some interesting developments since.

References

  1. 1.
    Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. Math. 160(2), 781–793 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ban, F., Jain, K., Papadimitriou, C.H., Psomas, C.A., Rubinstein, A.: Reductions in PPP. Manuscript (2016)Google Scholar
  3. 3.
    Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., Pitassi, T.: The relative complexity of NP search problems. In: 27th ACM Symposium on Theory of Computing, pp. 303–314 (1995)Google Scholar
  4. 4.
    Beckmann, A., Buss, S.: The NP Search Problems of Frege and Extended Frege Proofs, preliminary draft, December 2015Google Scholar
  5. 5.
    Bitansky, N., Paneth, O., Rosen, A.: On the cryptographic hardness of finding a Nash equilibrium. In: FOCS 2015, pp. 1480–1498 (2015)Google Scholar
  6. 6.
    Buss, S.: Bounded Arithmetic, Bibliopolis, Naples, Italy (1986). www.math.ucsd.edu/~sbuss/ResearchWeb/BAthesis/
  7. 7.
    Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. J. ACM 56(3), 1–57 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Daskalakis, C., Papadimitriou, C.H.: Continuous local search. In: SODA 2011, pp. 790–804 (2011)Google Scholar
  10. 10.
    Filos-Ratsikas, A., Frederiksen, S.K.S., Goldberg, P.W., Zhang, J.: Hardness Results for Consensus-Halving. Corr, arXiv:1609.05136 [cs.GT] (2016)
  11. 11.
    Garg, S., Pandey, O., Srinivasan, A.: Revisiting the cryptographic hardness of finding a Nash equilibrium. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 579–604. Springer, Heidelberg (2016). doi:10.1007/978-3-662-53008-5_20 CrossRefGoogle Scholar
  12. 12.
    Goldberg, P.W., Papadimitriou, C.H.: Towards a Unified Complexity Theory of Total Functions (2017). SubmittedGoogle Scholar
  13. 13.
    Herbrand, J.: Récherches sur la théorie de la démonstration. Ph.D. thesis, Université de Paris (1930)Google Scholar
  14. 14.
    Hubáček, P., Naor, M., Yogev, E.: The journey from NP to TFNP hardness. In: 8th ITCS (2017)Google Scholar
  15. 15.
    Jeřábek, E.: Integer factoring and modular square roots. J. Comput. Syst. Sci. 82(2), 380–394 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? J. Comput. Syst. Sci. 37(1), 79–100 (1988)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Komargodski, I., Naor, M., Yogev, E.: White-Box vs. Black-Box Complexity of Search Problems: Ramsey and Graph Property Testing. ECCC Report 15 (2017)Google Scholar
  18. 18.
    Krajíček, J.: Implicit proofs. J. Symbolic Logic 69(2), 387–397 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Megiddo, N.: A note on the complexity of \(P\)-matrix LCP and computing an equilibrium. Res. Rep. RJ6439, IBM Almaden Research Center, San Jose, pp. 1–5 (1988)Google Scholar
  20. 20.
    Megiddo, N., Papadimitriou, C.H.: On total functions, existence theorems and computational complexity. Theoret. Comput. Sci. 81(2), 317–324 (1991)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 498–532 (1994)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pudlák, P.: On the complexity of finding falsifying assignments for Herbrand disjunctions. Arch. Math. Logic 54, 769–783 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rubinstein, A.: Settling the complexity of computing approximate two-player Nash equilibria. In: FOCS (2016)Google Scholar
  24. 24.
    Varga, L.: Combinatorial Nullstellensatz modulo prime powers and the parity argument. arXiv:1402.4422 [math.CO] (2014)
  25. 25.
    Zhang, S.: Tight bounds for randomized and quantum local search. SIAM J. Comput. 39(3), 948–977 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of OxfordOxfordUK
  2. 2.University of California at BerkeleyBerkeleyUSA

Personalised recommendations