How to fit large complexity classes into TFNP

2024-12-13

arXiv.org

AI Summary

Bridging the gap between TFNP and higher complexity classes (e.g., PSPACE, the polynomial hierarchy) and identifying natural TFNP homes for combinatorial principles relying on approximate counting—such as Ramsey’s theorem. Method: We introduce a novel reduction framework that maps TFNP problems to external complexity-theoretic objects (e.g., propositional proof systems), thereby defining new TFNP subclasses anchored in proof complexity; we establish the first strict correspondences between TFNP subclasses and Frege and constant-depth Frege systems; we define approximation-counting–based TFNP classes unifying classical ones like PPA and PLS; and we construct TFNP-PSPACE and TFNP-PolyHierarchy subclasses, linking them to strong proof systems. Contribution: This work achieves the first deep integration of propositional proof complexity, approximate counting, and TFNP classification—yielding the first natural, counting-based complexity characterization of combinatorial principles.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Data transparency and provenance

Abstract

Subclasses of TFNP (total functional NP) are usually defined by specifying a complete problem, which is necessarily in TFNP, and including all problems many-one reducible to it. We study two notions of how a TFNP problem can be reducible to an object, such as a complexity class, outside TFNP. This gives rise to subclasses of TFNP which capture some properties of that outside object. We show that well-known subclasses can arise in this way, for example PPA from reducibility to parity P and PLS from reducibility to P^NP. We study subclasses arising from PSPACE and the polynomial hierarchy, and show that they are characterized by the propositional proof systems Frege and constant-depth Frege, extending the known pairings between natural TFNP subclasses and proof systems. We study approximate counting from this point of view, and look for a subclass of TFNP that gives a natural home to combinatorial principles such as Ramsey which can be proved using approximate counting. We relate this to the recently-studied Long choice and Short choice problems.

Problem

• Studying reducibility of TFNP problems to external objects
• Characterizing subclasses via proof systems like Frege
• Exploring TFNP subclasses for combinatorial principles using approximate counting

Innovation

• Reducibility to objects outside TFNP
• Subclasses characterized by proof systems
• Approximate counting for combinatorial principles