Downward self-reducibility in the total function polynomial hierarchy

TL;DR

This paper extends the understanding of downward self-reducibility from decision problems to search problems in higher levels of the polynomial hierarchy, revealing collapse phenomena and providing new complexity bounds.

Contribution

It generalizes the collapse results for downward self-reducibility to higher polynomial hierarchy levels and introduces new upper bounds for specific search problems.

Findings

Downward self-reducibility implies complexity class collapses in higher hierarchies.

Problems with randomized downward self-reduction are in PLS^{Σ_{i-1}^P}.

Range Avoidance and Linear Ordering Principle are in UEOPL^{NP}.

Abstract

A problem $\mathcal{P}$ is considered downward self-reducible, if there exists an efficient algorithm for $\mathcal{P}$ that is allowed to make queries to only strictly smaller instances of $\mathcal{P}$. Downward self-reducibility has been well studied in the case of decision problems, and it is well known that any downward self-reducible problem must lie in $\mathsf{PSPACE}$. Harsha, Mitropolsky and Rosen [ITCS, 2023] initiated the study of downward self reductions in the case of search problems. They showed the following interesting collapse: if a problem is in $\mathsf{TFNP}$ and also downward self-reducible, then it must be in $\mathsf{PLS}$. Moreover, if the problem admits a unique solution then it must be in $\mathsf{UEOPL}$. We demonstrate that this represents just the tip of a much more general phenomenon, which holds for even harder search problems that lie higher up in the total function polynomial hierarchy ($\mathsf{TF\Sigma_i^P}$). In fact, even if we allow our downward self-reduction to be much more powerful, such a collapse will still occur. We show that any problem in $\mathsf{TF\Sigma_i^P}$ which admits a randomized downward self-reduction with access to a $\mathsf{\Sigma_{i-1}^P}$ oracle must be in $\mathsf{PLS}^{\mathsf{\Sigma_{i-1}^P}}$. If the problem has \textit{essentially unique solutions} then it lies in $\mathsf{UEOPL}^{\mathsf{\Sigma_{i-1}^P}}$. As one (out of many) application of our framework, we get new upper bounds for the problems $\mathrm{Range Avoidance}$ and $\mathrm{Linear Ordering Principle}$ and show that they are both in $\mathsf{UEOPL}^{\mathsf{NP}}$.