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Total NP Search Problems with Abundant Solutions

Author Jiawei Li

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Author Details

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

Funding

  • Li, Jiawei: Supported by Scott Aaronson’s Vannevar Bush Fellowship from the US Department of Defense, the Berkeley NSF-QLCI CIQC Center, a Simons Investigator Award, and the Simons "It from Qubit" collaboration. Also funded in part by Igor Carboni Oliveira’s the Royal Society University Research Fellowship URFbackslash R1backslash 191059 and by the EPSRC New Horizons Grant EP/V048201/1.

Acknowledgements

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)
https://doi.org/10.4230/LIPIcs.ITCS.2024.75

Abstract

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
Keywords
  • TFNP
  • Pigeonhole Principle

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