Year: 2025
Type: article
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 $$i\in [n]$$ i ∈ [ n ] , the set $$E_i$$ E i of edges connecting vertices along dimension i can be decomposed into equivalence classes (so-called phases), such that flipping the direction of any $$S\subseteq E_i$$ S ⊆ E i 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 $$O(n\cdot 3^n)$$ O ( n · 3 n ) time, significantly improving upon the previously known $$O(n\cdot 4^n)$$ O ( n · 4 n ) 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 $$\textsf{PSPACE}$$ 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.
Source: Graphs and Combinatorics
Authors: Michaela Borzechowski, Simon Weber
Institutions: Freie Universität Berlin, ETH Zurich
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,190
OpenAlex ID: W4409796002