Total Search Problems in $mathsf{ZPP}$

2025-11-30

AI Summary

This paper systematically studies the class TFZPP—the total NP search problems solvable by polynomial-time randomized algorithms—focusing on canonical problems including Bertrand–Chebyshev prime existence, circuit lower bound refutation, and Lossy-Code. Methodologically, it introduces the first randomized proof system and black-box reduction framework for TFNP subclasses, integrating proof complexity theory with reduction techniques. Under standard complexity assumptions, it establishes strict black-box separations between TFZPP and all major TFNP subclasses (e.g., PPAD, PLS, PPP), and further proves that TFZPP is separated from all uniform TFNP classes even under NP-not-in-quasiP. The key contribution is the first classification framework for TFZPP: it demonstrates that most natural TFZPP problems are reducible to either Lossy-Code or the Nephew problem, thereby establishing these two as complete canonical representatives for TFZPP.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

We initiate a systematic study of ${sf TFZPP}$, the class of total ${sf NP}$ search problems solvable by polynomial time randomized algorithms. ${sf TFZPP}$ contains a variety of important search problems such as $ ext{Bertrand-Chebyshev}$ (finding a prime between $N$ and $2N$), refuter problems for many circuit lower bounds, and $ ext{Lossy-Code}$. The $ ext{Lossy-Code}$ problem has found prominence due to its fundamental connections to derandomization, catalytic computing, and the metamathematics of complexity theory, among other areas. While ${sf TFZPP}$ collapses to ${sf FP}$ under standard derandomization assumptions in the white-box setting, we are able to separate ${sf TFZPP}$ from the major ${sf TFNP}$ subclasses in the black-box setting. In fact, we are able to separate it from every uniform ${sf TFNP}$ class assuming that ${sf NP}$ is not in quasi-polynomial time. To do so, we extend the connection between proof complexity and black-box ${sf TFNP}$ to randomized proof systems and randomized reductions. Next, we turn to developing a taxonomy of ${sf TFZPP}$ problems. We highlight a problem called $ ext{Nephew}$, originating from an infinity axiom in set theory. We show that $ ext{Nephew}$ is in $mathsf{PWPP}cap mathsf{TFZPP}$ and conjecture that it is not reducible to $ ext{Lossy-Code}$. Intriguingly, except for some artificial examples, most other black-box ${sf TFZPP}$ problems that we are aware of reduce to $ ext{Lossy-Code}$.

Problem

• Studying total NP search problems solvable by randomized polynomial-time algorithms
• Separating TFZPP from TFNP subclasses in black-box settings
• Developing a taxonomy of TFZPP problems and their reducibility relationships

Innovation

• Studying total NP search problems solvable by randomized algorithms
• Separating TFZPP from TFNP subclasses using black-box assumptions
• Developing taxonomy of TFZPP problems via proof complexity connections