Neighborhood stability under non-negative cardinal preferences on path seat graphs

Determine whether neighborhood stable assignments always exist for path seat graphs when agents have non-negative cardinal preferences over neighbors, in contrast to the impossibility known for cycle seat graphs under the same preference model.

Background

The paper’s main positive results concern binary preferences: it provides polynomial-time algorithms ensuring neighborhood stability on cycles and paths. The authors note that extending to non-negative cardinal preferences leads to impossibility for cycles, indicating that the binary assumption is critical there.

However, it is explicitly stated that whether a similar impossibility holds—or whether neighborhood stable assignments do exist—on path seat graphs under non-negative cardinal preferences remains unresolved, making this a targeted open question.