Pseudo Unique Sink Orientations

TL;DR

This paper characterizes and counts pseudo unique sink orientations (PUSOs) in n-cubes, revealing their rigid structure and showing their count is negligible compared to the total USOs, with new classes of USOs introduced.

Contribution

It introduces and characterizes PUSOs, providing bounds on their number and defining new classes of USOs such as border and odd USOs.

Findings

PUSOs are structurally more rigid than USOs.

Number of PUSOs is negligible compared to USOs.

New classes of USOs, border and odd, are characterized.

Abstract

A unique sink orientation (USO) is an orientation of the $n$-dimensional cube graph ($n$-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is $n^{\Theta(2^n)}$. If a cube orientation is not a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn't. In this paper, we characterize and count PUSOs of the $n$-cube. We show that PUSOs have a much more rigid structure than USOs and that their number is between $2^{\Omega(2^{n-\log n})}$ and $2^{O(2^n)}$ which is negligible compared to the number of USOs. As tools, we introduce and characterize two new classes of USOs: border USOs (USOs that appear as facets of PUSOs), and odd USOs which are dual to border USOs but easier to understand.