Downward Self-Reducibility in TFNP

Authors Prahladh Harsha , Daniel Mitropolsky, Alon Rosen

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Author Details

Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India
Daniel Mitropolsky
  • Columbia University, New York, NY, USA
Alon Rosen
  • Bocconi University, Milano, Italy
  • Reichman University, Herzliya, Israel


This work was initiated when the first and second authors were visiting the third author at Bocconi University and we are thankful to Boccconi University for their hospitality. We thank Pavel Hubácek, Eylon Yogev, and Omer Paneth for their comments on an earlier draft of this paper. We also thank the anonymous referees for several useful comments and informing us of End-Of-Potential-Line and UniqueEOPL, the complete problems for the classes CLS and UEOPL respectively, which simplified certain parts of our proof.

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Prahladh Harsha, Daniel Mitropolsky, and Alon Rosen. Downward Self-Reducibility in TFNP. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A problem is downward self-reducible if it can be solved efficiently given an oracle that returns solutions for strictly smaller instances. In the decisional landscape, downward self-reducibility is well studied and it is known that all downward self-reducible problems are in PSPACE. In this paper, we initiate the study of downward self-reducible search problems which are guaranteed to have a solution - that is, the downward self-reducible problems in TFNP. We show that most natural PLS-complete problems are downward self-reducible and any downward self-reducible problem in TFNP is contained in PLS. Furthermore, if the downward self-reducible problem is in TFUP (i.e. it has a unique solution), then it is actually contained in UEOPL, a subclass of CLS. This implies that if integer factoring is downward self-reducible then it is in fact in UEOPL, suggesting that no efficient factoring algorithm exists using the factorization of smaller numbers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • downward self-reducibility
  • TFNP
  • TFUP
  • factoring
  • PLS
  • CLS


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