# On Flipping Edge Sets in Unique Sink Orientations

## TL;DR

This paper improves the efficiency of computing phases in unique sink orientations and explores their structural and computational properties.

## Contribution

An improved algorithm for computing phases in USOs and new insights into their computational complexity and structure.

## Key findings

- An algorithm to compute phases in USOs in O(n·3^n) time, improving on the previous O(n·4^n) method.
- Flipping a phase can be done in polynomial space relative to the USO storage.
- Determining if two edges are in the same phase is PSPACE-complete for succinctly encoded USOs.

## Abstract

A unique sink orientation (USO) is an orientation of the n-dimensional hypercube graph such that every non-empty face contains a unique sink. We consider the only known connected flip graph on USOs. This flip graph is based on the following theorem due to Schurr: given any n-dimensional USO and any one dimension \documentclass[12pt]{minimal}
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\begin{document}$$i\in [n]$$\end{document}i∈[n], the set \documentclass[12pt]{minimal}
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\begin{document}$$E_i$$\end{document}Ei of edges connecting vertices along dimension i can be decomposed into equivalence classes (so-called phases), such that flipping the direction of any \documentclass[12pt]{minimal}
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\begin{document}$$S\subseteq E_i$$\end{document}S⊆Ei yields another USO if and only if S is the union of some of these phases. In this paper we provide an algorithm to compute the phases of a given USO in \documentclass[12pt]{minimal}
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\begin{document}$$O(n\cdot 3^n)$$\end{document}O(n·3n) time, significantly improving upon the previously known \documentclass[12pt]{minimal}
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\begin{document}$$O(n\cdot 4^n)$$\end{document}O(n·4n) trivial algorithm. We also show that the phase containing a given edge can be flipped using only poly(n) space additional to the space required to store the USO. We contrast this by showing that given a boolean circuit of size poly(n) succinctly encoding an n-dimensional USO, it is \documentclass[12pt]{minimal}
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\begin{document}$$\textsf{PSPACE}$$\end{document}PSPACE-complete to determine whether two given edges are in the same phase. Finally, we also prove some new results on the structure of phases.