The hardness results for the maximum stable marriage problem with general partial orders highlight the increased complexity and difficulty in achieving efficient approximations when preferences are not strictly ordered. Specifically, the paper establishes that it is NP-hard to approximate the problem within a factor better than 2 under the Unique Games Conjecture (UGC). This result indicates that as the structure of preferences becomes more complex, the challenges in finding stable matchings also escalate.
This relationship extends to other stable matching problems that involve uncertain or changing preferences. For instance, in scenarios where agents have preferences that may evolve over time or are partially unknown, the problem of finding a stable matching becomes significantly more challenging. The presence of uncertainty can lead to situations where the preferences of agents are not only dynamic but also interdependent, complicating the stability conditions.
Moreover, the hardness results for general partial orders suggest that similar inapproximability results may hold for stable matching problems with uncertain preferences. If the preferences can be modeled as partial orders that change based on external factors or new information, the underlying complexity may mirror that of the maximum stable marriage problem with general partial orders. Thus, the findings in this paper provide a foundational understanding of the limitations and challenges faced in stable matching problems with complex preference structures, indicating that achieving efficient approximations may be inherently difficult in these contexts.