coNP membership of Uso-Completability

Determine whether the decision problem Uso-Completability—given a partial outmap on the vertices of an n-dimensional hypercube, decide if it extends to the outmap of a unique sink orientation—belongs to coNP when the partial outmap is represented by a Boolean circuit.

Background

The authors define Uso-Completability as the decision problem of determining whether a partial outmap can be completed to a full USO outmap. They note that Uso-Completability is at least as hard as Uso-Recognition (deciding if a complete outmap is a USO), which is coNP-hard when represented as a Boolean circuit.

Although clashes provide efficiently checkable certificates for Uso-Recognition, they do not suffice for Uso-Completability because some non-completable partial outmaps have no clashes. Consequently, it remains unresolved whether Uso-Completability lies in coNP.