Downward self-reducibility in the total function polynomial hierarchy

2025-07-25

AI Summary

This paper investigates downward self-reducibility of search problems in the total function polynomial hierarchy (TFΣᵢᴾ) and its induced complexity collapses. We introduce a randomized downward self-reduction framework equipped with a Σᵢ₋₁ᴾ oracle, enabling recursive solution of higher-order search problems within probabilistic polynomial time, and characterize the interplay between solution uniqueness and lower-level complexity classes such as PLS^{Σᵢ₋₁ᴾ} and UEOPL^{Σᵢ₋₁ᴾ}. Our key contributions are: (i) establishing that long-standing open problems—including Range Avoidance and the Linear Ordering Principle—belong to UEOPLᴺᴾ, thereby providing tight upper bounds; and (ii) developing the first general collapse mechanism applicable to higher-order TF classes, unifying structural explanations for hardness barriers across multiple canonical problems. This work extends TFNP-level collapse results to TFΣᵢᴾ, significantly advancing our understanding of the intrinsic structure and limitations of total function hierarchies.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance

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 ($mathsf{TFΣ_i^P}$). 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 $mathsf{TFΣ_i^P}$ which admits a randomized downward self-reduction with access to a $mathsf{Σ_{i-1}^P}$ oracle must be in $mathsf{PLS}^{mathsf{Σ_{i-1}^P}}$. If the problem has extit{essentially unique solutions} then it lies in $mathsf{UEOPL}^{mathsf{Σ_{i-1}^P}}$. 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 $mathsf{UEOPL}^{mathsf{NP}}$.

Problem

• Studies downward self-reducibility in total function polynomial hierarchy.
• Shows collapse of downward self-reducible problems into PLS classes.
• Provides new upper bounds for Range Avoidance and Linear Ordering.

Innovation

• Randomized downward self-reduction technique
• Oracle access to lower hierarchy levels
• Generalization to total function polynomial hierarchy