Downward Self-Reducibility in TFNP
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 -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.
Cite
@article{arxiv.2209.10509,
title = {Downward Self-Reducibility in TFNP},
author = {Prahladh Harsha and Daniel Mitropolsky and Alon Rosen},
journal= {arXiv preprint arXiv:2209.10509},
year = {2023}
}
Comments
16 pages, 2 figures