Extension of neighborhood stability results beyond binary preferences

Determine whether the existence guarantees for neighborhood stable assignments established in this paper under binary preferences extend to more general (non-binary) preference models, such as multi-valued or cardinal utilities over neighbors, for the seat graphs considered in the study (including cycles and paths).

Background

The paper proves that neighborhood stable assignments exist and can be computed in polynomial time for cycle and path seat graphs when agents have binary (dichotomous) preferences over neighbors. Beyond these settings, the authors show that neighborhood stable assignments are not guaranteed to exist in general, and they provide sufficient conditions involving the directed feedback vertex set number of the preference graph.

Given these results, a natural question is whether the positive existence guarantees persist once preferences are generalized beyond binary approvals. The authors explicitly note uncertainty about such extensions, highlighting that the scope of their main results may be limited to binary preferences and motivating further investigation into broader preference domains.