Total NP Search Problems with Abundant Solutions

Author Jiawei Li

Thumbnail PDF


  • Filesize: 0.9 MB
  • 23 pages

Document Identifiers

Author Details

Jiawei Li
  • The University of Texas at Austin, TX, USA


We thank the the anonymous ITCS reviewers for their helpful comments, especially for pointing out the relevant paper by Müller [Moritz Müller, 2021]. We thank Sid Jain, Robert Robere, Hanlin Ren, Yuhao Li, Rahul Santhanam, Shuichi Hirahara, and Scott Aaronson for discussions. We would also like to express our special thanks to Yurong Chen, Zhiyang Xun, and ChatGPT for their kind assistance in the writing of this paper.

Cite AsGet BibTex

Jiawei Li. Total NP Search Problems with Abundant Solutions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 75:1-75:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We define a new complexity class TFAP to capture TFNP problems that possess abundant solutions for each input. We identify several problems across diverse fields that belong to TFAP, including WeakPigeon (finding a collision in a mapping from [2n] pigeons to [n] holes), Yamakawa-Zhandry’s problem [Takashi Yamakawa and Mark Zhandry, 2022], and all problems in TFZPP. Conversely, we introduce the notion of "semi-gluability" to characterize TFNP problems that could have a unique or a very limited number of solutions for certain inputs. We prove that there is no black-box reduction from any "semi-gluable" problems to any TFAP problems. Furthermore, it can be extended to rule out randomized black-box reduction in most cases. We identify that the majority of common TFNP subclasses, including PPA, PPAD, PPADS, PPP, PLS, CLS, SOPL, and UEOPL, are "semi-gluable". This leads to a broad array of oracle separation results within TFNP regime. As a corollary, UEOPL^O ⊈ PWPP^O relative to an oracle O.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • TFNP
  • Pigeonhole Principle


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


  1. Scott Aaronson. Lower bounds for local search by quantum arguments. SIAM J. Comput., 35(4):804-824, 2006. URL:
  2. Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, and Avishay Tal. Degree vs. approximate degree and quantum implications of huang’s sensitivity theorem. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1330-1342. ACM, 2021. URL:
  3. Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595-605, 2004. URL:
  4. Paul Beame, Stephen A. Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The relative complexity of NP search problems. In Frank Thomson Leighton and Allan Borodin, editors, Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, 29 May-1 June 1995, Las Vegas, Nevada, USA, pages 303-314. ACM, 1995. URL:
  5. Simina Brânzei and Jiawei Li. The query complexity of local search and brouwer in rounds. In Po-Ling Loh and Maxim Raginsky, editors, Conference on Learning Theory, 2-5 July 2022, London, UK, volume 178 of Proceedings of Machine Learning Research, pages 5128-5145. PMLR, 2022. URL:
  6. Josh Buresh-Oppenheim and Tsuyoshi Morioka. Relativized NP search problems and propositional proof systems. In 19th Annual IEEE Conference on Computational Complexity (CCC 2004), 21-24 June 2004, Amherst, MA, USA, pages 54-67. IEEE Computer Society, 2004. URL:
  7. Samuel R. Buss, Leszek Aleksander Kolodziejczyk, and Neil Thapen. Fragments of approximate counting. J. Symb. Log., 79(2):496-525, 2014. URL:
  8. Xi Chen and Xiaotie Deng. Matching algorithmic bounds for finding a brouwer fixed point. J. ACM, 55(3):13:1-13:26, 2008. URL:
  9. Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM (JACM), 56(3):14, 2009. Google Scholar
  10. Xi Chen, Yuhao Li, and Mihalis Yannakakis. Reducing tarski to unique tarski (in the black-box model). In Amnon Ta-Shma, editor, 38th Computational Complexity Conference, CCC 2023, July 17-20, 2023, Warwick, UK, volume 264 of LIPIcs, pages 21:1-21:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL:
  11. Xi Chen, Xiaoming Sun, and Shang-Hua Teng. Quantum separation of local search and fixed point computation. Algorithmica, 56(3):364-382, 2010. URL:
  12. Constantinos Daskalakis, Paul W Goldberg, and Christos H Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009. Google Scholar
  13. Constantinos Daskalakis and Christos Papadimitriou. Continuous local search. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms, pages 790-804. SIAM, 2011. Google Scholar
  14. Constantinos Daskalakis, Stratis Skoulakis, and Manolis Zampetakis. The complexity of constrained min-max optimization. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1466-1478. ACM, 2021. URL:
  15. Susanna F. de Rezende, Mika Göös, and Robert Robere. Guest column: Proofs, circuits, and communication. SIGACT News, 53(1):59-82, 2022. URL:
  16. John Fearnley, Paul Goldberg, Alexandros Hollender, and Rahul Savani. The complexity of gradient descent: CLS = PPAD ∩ PLS. J. ACM, 70(1):7:1-7:74, 2023. URL:
  17. John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique end of potential line. J. Comput. Syst. Sci., 114:1-35, 2020. URL:
  18. Aris Filos-Ratsikas and Paul W Goldberg. The complexity of splitting necklaces and bisecting ham sandwiches. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 638-649, 2019. Google Scholar
  19. Paul W Goldberg and Alexandros Hollender. The hairy ball problem is ppad-complete. arXiv preprint, 2019. URL:
  20. Oded Goldreich, Shafi Goldwasser, and Dana Ron. On the possibilities and limitations of pseudodeterministic algorithms. In Robert D. Kleinberg, editor, Innovations in Theoretical Computer Science, ITCS '13, Berkeley, CA, USA, January 9-12, 2013, pages 127-138. ACM, 2013. URL:
  21. Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Further collapses in TFNP. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 33:1-33:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  22. Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Separations in proof complexity and TFNP. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 1150-1161. IEEE, 2022. URL:
  23. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, volume 124 of LIPIcs, pages 38:1-38:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  24. Mika Göös and Aviad Rubinstein. Near-optimal communication lower bounds for approximate nash equilibria. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 397-403. IEEE Computer Society, 2018. URL:
  25. Pavel Hubácek, Chethan Kamath, Karel Král, and Veronika Slívová. On average-case hardness in TFNP from one-way functions. In Rafael Pass and Krzysztof Pietrzak, editors, Theory of Cryptography - 18th International Conference, TCC 2020, Durham, NC, USA, November 16-19, 2020, Proceedings, Part III, volume 12552 of Lecture Notes in Computer Science, pages 614-638. Springer, 2020. URL:
  26. Pavel Hubácek and Eylon Yogev. Hardness of continuous local search: Query complexity and cryptographic lower bounds. SIAM J. Comput., 49(6):1128-1172, 2020. URL:
  27. Siddhartha Jain, Jiawei Li, Robert Robere, and Zhiyang Xun. On Pigeonhole Principles and Ramsey in TFNP. In submission, 2023. Google Scholar
  28. Emil Jeřábek. Integer factoring and modular square roots. Journal of Computer and System Sciences, 82(2):380-394, 2016. Google Scholar
  29. David S Johnson, Christos H Papadimitriou, and Mihalis Yannakakis. How easy is local search? Journal of computer and system sciences, 37(1):79-100, 1988. Google Scholar
  30. Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos H. Papadimitriou. Total functions in the polynomial hierarchy. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 44:1-44:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  31. Veronika Králová. On the complexity of search problems with a unique solution. Charles University Digital Repository, 2021. Google Scholar
  32. Nimrod Megiddo and Christos H Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81(2):317-324, 1991. Google Scholar
  33. Tsuyoshi Morioka. Classification of search problems and their definability in bounded arithmetic. Electron. Colloquium Comput. Complex., TR01-082, 2001. URL:
  34. Moritz Müller. Typical forcings, NP search problems and an extension of a theorem of riis. Ann. Pure Appl. Log., 172(4):102930, 2021. URL:
  35. Christos H Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and system Sciences, 48(3):498-532, 1994. Google Scholar
  36. Amol Pasarkar, Christos H. Papadimitriou, and Mihalis Yannakakis. Extremal combinatorics, iterated pigeonhole arguments and generalizations of PPP. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference, ITCS 2023, January 10-13, 2023, MIT, Cambridge, Massachusetts, USA, volume 251 of LIPIcs, pages 88:1-88:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL:
  37. Pavel Pudlák. On the complexity of finding falsifying assignments for herbrand disjunctions. Arch. Math. Log., 54(7-8):769-783, 2015. URL:
  38. Alon Rosen, Gil Segev, and Ido Shahaf. Can ppad hardness be based on standard cryptographic assumptions? In Theory of Cryptography Conference, pages 747-776. Springer, 2017. Google Scholar
  39. Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev., 41(2):303-332, 1999. URL:
  40. Daniel R. Simon. Finding collisions on a one-way street: Can secure hash functions be based on general assumptions? In Kaisa Nyberg, editor, Advances in Cryptology - EUROCRYPT '98, International Conference on the Theory and Application of Cryptographic Techniques, Espoo, Finland, May 31 - June 4, 1998, Proceeding, volume 1403 of Lecture Notes in Computer Science, pages 334-345. Springer, 1998. URL:
  41. Katerina Sotiraki, Manolis Zampetakis, and Giorgos Zirdelis. Ppp-completeness with connections to cryptography. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 148-158. IEEE, 2018. Google Scholar
  42. Salil P. Vadhan. Pseudorandomness. Found. Trends Theor. Comput. Sci., 7(1-3):1-336, 2012. URL:
  43. Takashi Yamakawa and Mark Zhandry. Verifiable quantum advantage without structure. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 69-74. IEEE, 2022. URL:
  44. Shengyu Zhang. Tight bounds for randomized and quantum local search. SIAM J. Comput., 39(3):948-977, 2009. URL: