On Pigeonhole Principles and Ramsey in TFNP

TL;DR

This work provides a decisive black-box separation between Ramsey in TFNP and the Pigeonhole Principle, refuting the conjecture that RAMSEY could be reduced to PPP in the black-box setting. It introduces a generalized pigeonhole framework, the t-Pigeon family, and the Pecking Order—a hierarchy of TFNP classes defined by black-box reductions to t-Pigeon variants, with polynomial vs subpolynomial growth in the collision parameter delineating PAP and SAP. The authors develop generalized collision-free pseudoexpectations and gluability concepts to prove lower bounds and separations, establishing a near-complete map of how Ramsey, BiRamsey, PLC, UPLC, PLS, and PPA relate within the Pecking Order. They also connect these results to cryptographic notions (PWPP/MCRH) and to quantum complexity via the Yamakawa-Zhandry problem, highlighting both the power and limits of black-box reductions and pointing to rich open problems at the interface of extremal combinatorics, proof complexity, and cryptography.

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.

Figures (2)

• Figure 1: Complexity classes and problems defined by Generalized Pigeonhole Principles and Ramsey. An arrow ${\text{\upshape\sffamily A}}\xspace\rightarrow{\text{\upshape\sffamily B}}\xspace$ means ${\text{\upshape\sffamily A}}\xspace\subseteq {\text{\upshape\sffamily B}}\xspace$. An orange arrow ${\text{\upshape\sffamily A}}\xspace~{\color{YellowOrange} \rightarrow} ~{\text{\upshape\sffamily B}}\xspace$ means ${\text{\upshape\sffamily A}}\xspace \subseteq {\text{\upshape\sffamily B}}\xspace$ and ${\text{\upshape\sffamily B}}\xspace \not\subseteq {\text{\upshape\sffamily A}}\xspace$ in the black-box setting. A dashed arrow ${\text{\upshape\sffamily A}}\xspace\dashrightarrow{\text{\upshape\sffamily B}}\xspace$ means ${\text{\upshape\sffamily A}}\xspace\not\subseteq {\text{\upshape\sffamily B}}\xspace$ in the black-box setting. All black-box separations in this figure are contributions of this work, labelled by the corresponding theorem. We refer to the tower of $t$- PPP classes as the Pecking Order, while the union of $t$- PPP for all constant $t$ is referred to as the Pigeon Hierarchy ( PiH). We note that while our result ${\text{\upshape\scshape BiRamsey}}\xspace \not\in {\text{\upshape\sffamily SAP}}\xspace$ (in the black-box setting) applies for the standard parameter regime, ${\text{\upshape\scshape BiRamsey}}\xspace \in {\text{\upshape\sffamily PAP}}\xspace$ is for a slightly smaller parameter.
• Figure 2: UPLC solutions and reductions. (a) illustrates the lower-triangular condition of solutions to UPLC. (b) is an illustration of \ref{['lem:pwpp-uplc']}. The blue shaded region is a valid solution to $n/2$- PWPP which is a subset of the yellow shaded region, the lower-triangular collision returned by UPLC. (c) is an illustration of \ref{['lem:uplc-pap']}. The blue shaded region is the collision returned by $n\textup{-Pigeon}$, and the yellow shaded region is computed by solving a $\log n$ size UPLC instance by brute force.

Theorems & Definitions (107)

• Conjecture 1: Goldberg & Papadimitriou GP17
• Theorem 1.1
• Definition 1.1
• Theorem 1.2
• Definition 1.3: $t$- PPP and $t$- PWPP
• Definition 1.4: Pigeon Hierarchy
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• Definition 1.7