In Polyamorous Scheduling, we are given an edge-weighted graph and must find a periodic schedule of matchings in this graph which minimizes the maximal weighted waiting time between consecutive occurrences of the same edge. This NP-hard problem generalises Bamboo Garden Trimming and is motivated by the need to find schedules of pairwise meetings in a complex social group. We present two different analyses of an approximation algorithm based on the Reduce-Fastest heuristic, from which we obtain first a 6-approximation and then a 5.24-approximation for Polyamorous Scheduling. We also strengthen the extant proof that there is no polynomial-time $(1+\delta)$-approximation algorithm for the Optimisation Polyamorous Scheduling problem for any $\delta < \frac1{12}$ unless P = NP to the bipartite case. The decision version of Polyamorous Scheduling has a notion of density, similar to that of Pinwheel Scheduling, where problems with density below the threshold are guaranteed to admit a schedule (cf. the recently proven 5/6 conjecture, Kawamura, STOC 2024). We establish the existence of a similar threshold for Polyamorous Scheduling and give the first non-trivial bounds on the poly density threshold.

翻译：在多配偶调度问题中，给定一个边加权图，需要在该图中找到一个周期性的匹配调度，以最小化同一条边在连续出现之间的最大加权等待时间。这个NP难问题推广了竹子园修剪问题，其动机在于需要在复杂社交群体中寻找成对会面的调度方案。我们针对基于Reduce-Fastest启发式算法的近似算法提出了两种不同的分析，由此首次为多配偶调度问题分别得到6-近似和5.24-近似解。同时，我们强化了现有证明：对于任意δ < 1/12，除非P = NP，否则优化多配偶调度问题不存在多项式时间的(1+δ)-近似算法，并将该结论推广至二分图情形。多配偶调度问题的判定版本具有类似风车调度的密度概念，其中密度低于阈值的实例保证存在可行调度（参见近期证明的5/6猜想，Kawamura, STOC 2024）。我们证明了多配偶调度问题存在类似阈值，并首次给出了多密度阈值的非平凡边界。