A Universal Construction for Unique Sink Orientations

TL;DR

This paper introduces a universal construction framework for all Unique Sink Orientations (USOs) of cubes, enabling systematic generation and analysis of USOs through generalized rewriting rules, including new transformations like partial and phase swaps.

Contribution

The paper develops a generalized rewriting rules framework that can construct any USO from lower-dimensional USOs, unifying and extending previous methods.

Findings

Rewriting rules can generate all USOs from lower-dimensional cases.

A new elementary transformation called partial swap is introduced.

Partial swaps are related to phase flips and are generalized to phase swaps.

Abstract

Unique Sink Orientations (USOs) of cubes can be used to capture the combinatorial structure of many essential algebraic and geometric problems. For various structural and algorithmic questions, including enumeration of USOs and algorithm analysis, it is crucial to have systematic constructions of USOs. While some construction methods for USOs already exist, each one of them has some significant downside. Most of the construction methods have limited expressivity -- USOs with some desired properties cannot be constructed. In contrast, the phase flips of Schurr can construct all USOs, but the operation is not well understood. We were inspired by techniques from cube tilings of space; we expand upon existing techniques in the area to develop generalized rewriting rules for USOs. These rewriting rules are a new construction framework which can be applied to all USOs. The rewriting rules can generate every USO using only USOs of lower dimension. The effect of any specific rewriting rule on an USO is simple to understand. A special case of our construction produces a new elementary transformation of USOs, which we call a partial swap. We further investigate the relationship between partial swaps and phase flips and generalize partial swaps to phase swaps.