We study a public event scheduling problem, where multiple public events are scheduled to coordinate the availability of multiple agents. The availability of each agent is determined by solving a separate flexible interval job scheduling problem, where the jobs are required to be preemptively processed. The agents want to attend as many events as possible, and their agreements are considered to be the total length of time during which they can attend these events. The goal is to find a schedule for events as well as the job schedule for each agent such that the total agreement is maximized. We first show that the problem is NP-hard, and then prove that a simple greedy algorithm achieves $\frac{1}{2}$-approximation when the whole timeline is polynomially bounded. Our method also implies a $(1-\frac{1}{e})$-approximate algorithm for this case. Subsequently, for the general timeline case, we present an algorithmic framework that extends a $\frac{1}{\alpha}$-approximate algorithm for the one-event instance to the general case that achieves $\frac{1}{\alpha+1}$-approximation. Finally, we give a polynomial time algorithm that solves the one-event instance, and this implies a $\frac{1}{2}$-approximate algorithm for the general case.

翻译：我们研究了一个公共事件调度问题，其中多个公共事件需协调安排，以契合多个智能体的可用时间。每个智能体的可用性由其独立求解一个灵活区间作业调度问题确定，该问题要求作业可抢占式处理。智能体希望参加尽可能多的事件，其参与度定义为能够参加这些事件的总时长。目标是找到事件调度方案及每个智能体的作业调度方案，使得总参与度最大化。我们首先证明该问题是NP难的，随后证明在时间轴多项式有界的情况下，一个简单贪心算法可达到$\frac{1}{2}$近似比。我们的方法在该情形下还隐含一个$(1-\frac{1}{e})$近似算法。进一步，针对一般时间轴情形，我们提出一个算法框架，将单事件实例的$\frac{1}{\alpha}$近似算法扩展为一般情形下达到$\frac{1}{\alpha+1}$近似比的算法。最后，我们给出一个多项式时间算法求解单事件实例，并由此得到一般情形的$\frac{1}{2}$近似算法。