On Pigeonhole Principles and Ramsey in TFNP

TL;DR

This paper demonstrates that the TFNP problem RAMSEY cannot be reduced to PIGEON in the black-box setting, introduces a hierarchy of generalized pigeonhole principles, and establishes new complexity separations and lower bounds.

Contribution

It introduces a hierarchy of generalized pigeonhole principles within TFNP, proves non-reducibility of RAMSEY to PIGEON, and develops new proof complexity lower bounds.

Findings

RAMSEY is not reducible to PIGEON in the black-box setting

Hierarchy of generalized pigeonhole classes t-PPP is established

New lower bounds in propositional proof complexity based on pseudoexpectation operators

Abstract

We show that the TFNP problem RAMSEY is not black-box reducible to PIGEON, refuting a conjecture of Goldberg and Papadimitriou in the black-box setting. We prove this by giving reductions to RAMSEY from a new family of TFNP problems that correspond to generalized versions of the pigeonhole principle, and then proving that these generalized versions cannot be reduced to PIGEON. Formally, we define t-PPP as the class of total NP-search problems reducible to finding a t-collision in a mapping from (t-1)N+1 pigeons to N holes. These classes are closely related to multi-collision resistant hash functions in cryptography. We show that the generalized pigeonhole classes form a hierarchy as t increases, and also give a natural condition on the parameters t1, t2 that captures exactly when t1-PPP and t2-PPP collapse in the black-box setting. Finally, we prove other inclusion and separation results between these generalized PIGEON problems and other previously studied TFNP subclasses, such as PLS, PPA and PLC. Our separation results rely on new lower bounds in propositional proof complexity based on pseudoexpectation operators, which may be of independent interest.