On Phases of Unique Sink Orientations

TL;DR

This paper investigates the structure of phases in unique sink orientations of hypercubes, providing an efficient algorithm for computing all phases and establishing the PSPACE-completeness of determining phase membership from a circuit encoding.

Contribution

It introduces a polynomial-time algorithm to compute all phases of a USO and proves the PSPACE-completeness of phase membership decision from circuit encodings.

Findings

All phases can be computed in $O(3^n)$ time.

Determining if two edges are in the same phase is PSPACE-complete.

The structure of phases is characterized and analyzed.

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. Schurr showed that given any $n$-dimensional USO and any dimension $i$, the set of edges $E_i$ in that dimension can be decomposed into equivalence classes (so-called phases), such that flipping the orientation of a subset $S$ of $E_i$ yields another USO if and only if $S$ is a union of a set of these phases. In this paper we prove various results on the structure of phases. Using these results, we show that all phases can be computed in $O(3^n)$ time, significantly improving upon the previously known $O(4^n)$ trivial algorithm. Furthermore, we show that given a boolean circuit of size $poly(n)$ succinctly encoding an $n$-dimensional (acyclic) USO, it is PSPACE-complete to determine whether two given edges are in the same phase. The problem is thus equally difficult as determining whether the hypercube orientation encoded by a given circuit is an acyclic USO [G\"artner and Thomas, STACS'15].