Further Collapses in TFNP

Abstract

We show $\text{EOPL}=\text{PLS}\cap \text{PPAD}$. Here the class $\text{EOPL}$ 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 $\text{CLS}=\text{PLS}\cap \text{PPAD}$ by Fearnley et al. (STOC 2021). We also prove a companion result $\text{SOPL}=\text{PLS}\cap \text{PPADS}$, where $\text{SOPL}$ is the class associated with the Sink-of-Potential-Line problem.