Finding schedules for pairwise meetings between the members of a complex social group without creating interpersonal conflict is challenging, especially when different relationships have different needs. We formally define and study the underlying optimisation problem: Polyamorous Scheduling. In Polyamorous Scheduling, we are given an edge-weighted graph and try to find a periodic schedule of matchings in this graph such that the maximal weighted waiting time between consecutive occurrences of the same edge is minimised. We show that the problem is NP-hard and that there is no efficient approximation algorithm with a better ratio than 13/12 unless P = NP. On the positive side, we obtain an $O(\log n)$-approximation algorithm. We also define a generalisation of density from the Pinwheel Scheduling Problem, "poly density", and ask whether there exists a poly density threshold similar to the 5/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024]. Polyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with respect to its optimisation variant, Bamboo Garden Trimming. Our work contributes the first nontrivial hardness-of-approximation reduction for any periodic scheduling problem, and opens up numerous avenues for further study of Polyamorous Scheduling.

翻译：为复杂社交群体中的成员安排成对会面时间表，同时避免人际冲突具有挑战性，尤其当不同关系存在不同需求时。我们正式定义并研究了其底层优化问题：多配偶调度（Polyamorous Scheduling）。在该问题中，给定一个边赋权图，我们需寻找该图中匹配的周期性调度方案，使得同一条边连续出现之间的最大加权等待时间最小化。我们证明该问题是NP难的，且除非P=NP，否则不存在近似比优于13/12的有效近似算法。从正面角度，我们获得了$O(\log n)$-近似算法。我们还定义了"波尔密度"（poly density）——这源于风车调度问题（Pinwheel Scheduling Problem）中密度概念的推广，并探究是否存在类似风车调度中5/6密度阈值[Kawamura, STOC 2024]的波尔密度阈值。多配偶调度是风车调度在其优化变体"竹园修剪"（Bamboo Garden Trimming）意义下的自然推广。我们的工作为任意周期性调度问题首次提供了非平凡的难近似性归约，并为多配偶调度的进一步研究开辟了众多方向。