Downward Self-Reducibility in TFNP

Prahladh Harsha; Daniel Mitropolsky; Alon Rosen

doi:10.4230/lipics.itcs.2023.67

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arXiv · 2209.10509

Downward Self-Reducibility in TFNP

Prahladh Harsha·Daniel Mitropolsky·Alon Rosen

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Abstract

A problem is \emph{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 \textsc{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 \textsc{TFNP}. We show that most natural $\PLS$-complete problems are downward self-reducible and any downward self-reducible problem in \textsc{TFNP} is contained in \textsc{PLS}. Furthermore, if the downward self-reducible problem is in \textsc{TFUP} (i.e. it has a unique solution), then it is actually contained in \textsc{UEOPL}, a subclass of \textsc{CLS}. This implies that if integer factoring is \emph{downward self-reducible} then it is in fact in \textsc{UEOPL}, suggesting that no efficient factoring algorithm exists using the factorization of smaller numbers.

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BibTeX RIS

Prahladh Harsha, Daniel Mitropolsky, Alon Rosen. 2022-09-21. Downward Self-Reducibility in TFNP. https://doi.org/10.4230/lipics.itcs.2023.67

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