Hierarchies within TFNP: building blocks and collapses

2025-07-29

AI Summary

This paper investigates the relative complexity among subclasses of TFNP, addressing the lack of a formal relativization framework for total search problems. Method: We introduce a novel oracle model and reduction framework tailored to TFNP, where oracles encode multi-solution search problems; define relativized composite classes (e.g., PPP^PPP, PPAD^PPA); construct complete problems for these classes; and conduct hierarchical and collapse analyses. Contribution/Results: We establish self-low properties for key TFNP subclasses—most notably proving PPA^PPA = PPA and analogous results for PLS—demonstrating their intrinsic stability under relativization. Furthermore, we lay the theoretical foundations for relativized reasoning across TFNP subclasses, enabling fine-grained classification of search problems. Our framework provides new tools and a systematic methodology for analyzing structural relationships within TFNP, thereby advancing the broader theory of total search problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Other Foundations of Search & Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We initiate the study of complexity classes $mathsf{A^B}$ where $mathsf{A}$ and $mathsf{B}$ are both $mathsf{TFNP}$ subclasses. For example, we consider complexity classes of the form $mathsf{PPP^{PPP}}$, $mathsf{PPAD^{PPA}}$, and $mathsf{PPA^{PLS}}$. We define complete problems for such classes, and show that they belong in $mathsf{TFNP}$. These definitions require some care, since unlike a class like $mathsf{PPA^{NP}}$, where the $mathsf{NP}$ oracle defines a function, in $mathsf{PPA^{PPP}}$, the oracle is for a search problem with many possible solutions. Intuitively, the definitions we introduce quantify over all possible instantiations of the $mathsf{PPP}$ oracle. With these notions in place, we then show that several $mathsf{TFNP}$ subclasses are self-low. In particular, $mathsf{PPA^{PPA}} = mathsf{PPA}$, $mathsf{PLS^{PLS}} = mathsf{PLS}$, and $mathsf{LOSSY^{LOSSY}} = mathsf{LOSSY}$. These ideas introduce a novel approach for classifying computational problems within $mathsf{TFNP}$, such as the problem of deterministically generating large primes.

Problem

• Study complexity classes of TFNP subclasses combinations
• Define complete problems for TFNP subclass combinations
• Show self-lowness properties of several TFNP subclasses

Innovation

• Defines complexity classes with TFNP subclasses
• Introduces complete problems for TFNP classes
• Shows self-low properties of TFNP subclasses