多线程调度
Polyamorous Scheduling
摘要：
在一个复杂的社会群体中，要找到一个不产生人际冲突的成对会议的时间表是很有挑战性的，特别是当不同的关系有不同的需求时。我们正式定义和研究的基本优化问题：多线程调度。在多线程调度中，我们给出了一个边加权图，并试图在这个图中找到一个周期性的匹配时间表，使得相同边的连续出现之间的最大加权等待时间最小化。我们表明，该问题是NP-难的，没有有效的近似算法比4/3，除非P = NP。在积极的一面，我们得到了一个$O(\log n)$ -近似算法;实际上，一个$O(\log Δ)$ -近似$Δ$的最大程度，即，是所有人中拥有最多关系的我们还从风车调度问题中定义了一个广义的密度，“多边形密度”，并询问是否存在一个类似于风车调度的5/6密度阈值的多边形密度阈值[Kawamura，STOC 2024]。多功能调度是风车调度的一个自然概括，相对于其优化变体，竹园修剪。我们的工作贡献了第一个非平凡的硬度的近似减少任何周期性调度问题，并开辟了许多途径，为进一步研究的多线程调度。
英文摘要：
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 4/3 unless P = NP. On the positive side, we obtain an $O(\log n)$-approximation algorithm; indeed, a $O(\log Δ)$-approximation for $Δ$ the maximum degree, i.e., the largest number of relationships of any individual. 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.
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