Non-Promise Version of Unique Sink Orientations

TL;DR

This paper investigates a non-promise version of the Unique Sink Orientation problem, aiming to find a sink or verify violations, and explores its complexity and algorithmic properties beyond the promise setting.

Contribution

It extends known algorithms and properties from the promise to the non-promise USO problem, analyzes its complexity class, and categorizes verifiable violations.

Findings

The non-promise USO problem is in the class PLS^dt.

Known algorithms like Fibonacci Seesaw are adapted for the non-promise case.

Violations with up to 4 vertices are fully categorized and efficiently verifiable.

Abstract

A unique sink orientation (USO) is an orientation of the edges of a hypercube such that each face has a unique sink. Many optimization problems like linear programs reduce to USOs, in the sense that each vertex corresponds to a possible solution, and the global sink corresponds to the optimal solution. People have been studying intensively the problem of find the sink of a USO using vertex evaluations, i.e., queries which return the orientation of the edges around a vertex. This problem is a so called promise problem, as it assumes that the orientation it receives is a USO. In this paper, we analyze a non-promise version of the USO problem, in which we try to either find a sink or an efficiently verifiable violation of the USO property. This problem is worth investigating, because some problems which reduce to USO are also promise problems (and so we can also define a non-promise version for them), and it would be interesting to discover where USO lies in the hierarchy of subclasses of $\texttt{TFNP}^\texttt{dt}$, and for this a total search problem is required (which is the case for the non-promise version). We adapt many known properties and algorithms from the promise version to the non-promise one, including known algorithms for small dimensions and lower and upper bounds, like the Fibonacci Seesaw Algorithm. Furthermore, we present an efficient resolution proof of the problem, which shows it is in the search complexity class $\texttt{PLS}^\texttt{dt}$ (although this fact was already known via reductions). Finally, although initially the only allowed violations consist of $2$ vertices, we generalize them to more vertices, and provide a full categorization of violations with $4$ vertices, showing that they are also efficiently verifiable.