How to fit large complexity classes into TFNP

TL;DR

This paper explores how subclasses of TFNP can be characterized by reducibility to complexity classes outside TFNP, linking them to proof systems and combinatorial principles, and extends the understanding of TFNP's structure.

Contribution

It introduces a framework for defining TFNP subclasses via reducibility to outside objects, connecting well-known subclasses to proof systems and combinatorial principles.

Findings

Characterization of PPA and PLS via reducibility to outside classes.

Extension of TFNP subclasses to PSPACE and the polynomial hierarchy.

Identification of a TFNP subclass related to approximate counting and combinatorial principles.

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.