Further Collapses in TFNP

TL;DR

This paper proves that the complexity class EOPL equals the intersection of PLS and PPAD, simplifying the understanding of total search problems and providing new insights into the collapse of related classes.

Contribution

It establishes the equality EOPL=PLS∩PPAD and offers a simpler proof of the collapse CLS=PLS∩PPAD, advancing the theoretical understanding of total search problem classes.

Findings

EOPL equals PLS intersect PPAD

Simplified proof of CLS collapse

Introduces the class SOPL and its relation to PLS and PPADS

Abstract

We show $\textsf{EOPL}=\textsf{PLS}\cap\textsf{PPAD}$. Here the class $\textsf{EOPL}$ consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubacek and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse $\textsf{CLS}=\textsf{PLS}\cap\textsf{PPAD}$ by Fearnley et al. (STOC 2021). We also prove a companion result $\textsf{SOPL}=\textsf{PLS}\cap\textsf{PPADS}$, where $\textsf{SOPL}$ is the class associated with the Sink-of-Potential-Line problem.