Unique Sink Orientations of Grids is in Unique End of Potential Line

TL;DR

This paper establishes that the problem of finding a sink in a grid with a unique sink orientation is in the complexity class UEOPL, linking it to a broader class of total search problems with unique solutions.

Contribution

It provides a promise-preserving reduction from GridUSO to a UEOPL-complete problem, demonstrating GridUSO's membership in UEOPL and its promise version in PUEOPL.

Findings

GridUSO is in the complexity class UEOPL.

Promise version of GridUSO is in PUEOPL.

Reduction preserves promises and solutions.

Abstract

The complexity classes Unique End of Potential Line (UEOPL) and its promise version PUEOPL were introduced in 2018 by Fearnly et al. UEOPL captures search problems where the instances are promised to have a unique solution. UEOPL captures total search versions of these promise problems. The promise problems can be made total by defining violations that are returned as a short certificate of an unfulfilled promise. GridUSO is the problem of finding the sink in a grid with a unique sink orientation. It was introduced by G\"artner et al. We describe a promise preserving reduction from GridUSO to UniqueForwardEOPL, a UEOPL-complete problem. Thus, we show that GridUSO is in UEOPL and its promise version is in PUEOPL.