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.

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.

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