最优难民安置的参数化算法
Parameterized Algorithms for Optimal Refugee Resettlement
AI总结：
针对带服务配额约束的最优难民安置问题，研究其可行分配、帕累托最优分配、最大总效用分配三类变体的计算复杂性，给出多类自然参数的固定参数可处理算法及对应的难处理结果。
摘要：
我们研究最优难民安置问题的若干变体，该问题需将一组难民家庭集合$F$以可行且最优的方式分配至一组可选安置地点集合$L$。可行性约束源于以下设定：每个家庭都需要特定服务（如住宿、学校名额或医疗援助），而每个给定地点提供的服务单元数量存在上限，且可能存在下限。除了寻找可行分配的问题外，我们还研究了两类自然的优化变体。第一类变体中，我们允许家庭表达对集合$P$中地点的偏好，目标是得到帕累托最优分配。在更通用的设定中，家庭可以为集合$P$中的每个地点赋予效用值，任务是找到总效用最大的可行分配。我们利用参数化复杂性框架，以多变量方式研究了所有三类变体的计算复杂性。我们针对若干自然参数化方案给出了固定参数可处理算法，并通过严格的难处理结果补充了这些可处理情形。
英文摘要：
We study variants of the Optimal Refugee Resettlement problem where a set $F$ of refugee families need to be allocated to a set $L$ of possible places of resettlement in a feasible and optimal way. Feasibility issues emerge from the assumption that each family requires certain services (such as accommodation, school seats, or medical assistance), while there is an upper and, possibly, a lower quota on the number of service units provided at a given place. Besides studying the problem of finding a feasible assignment, we also investigate two natural optimization variants. In the first one, we allow families to express preferences over $P$, and we aim for a Pareto-optimal assignment. In a more general setting, families can attribute utilities to each place in $P$, and the task is to find a feasible assignment with maximum total utilities. We study the computational complexity of all three variants in a multivariate fashion using the framework of parameterized complexity. We provide fixed-parameter tractable algorithms for a handful of natural parameterizations, and complement these tractable cases with tight intractability results.
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