AI 中文总结
该研究将公平分配与区间调度结合，针对离线和在线场景分别提出算法，证明了公平价格的理论下界，实验显示Greedy-Balanced算法表现优于最坏情况保证。
AI 中文摘要
我们从公平分配的角度研究区间调度问题。假设有m台相同的机器和一组区间，每个区间由开始时间、结束时间和非负权重指定。调度方案将区间子集分配给机器，使得同一台机器上的任意两个区间不重叠，目标是最大化已调度区间的总权重。将机器视为智能体，区间视为物品，我们要求调度方案满足无妒忌至多一个物品（EF1）的条件，并以无公平约束的离线最优解为基准衡量效率。在离线场景中，我们提出一种算法，该算法在无权区间情况下可计算出EF1调度方案，其效率损失至多为3/2倍；我们还证明了在无权区间和单位长度加权区间两种场景下，效率损失下界为(3m-2)/(2m-1)，当m趋近于无穷大时该下界趋近于3/2，因此极限情况下的公平价格为3/2。在在线场景中，区间按开始时间非递减顺序到达；到达的区间必须被接受或拒绝，拒绝不可撤销，且已接受的区间可在其结束前任意时间被撤销并丢失。对于无权区间场景，我们提出了Greedy-Balanced算法，这是一种简单的算法，可在每个时间点维持EF1属性，且相对于无公平约束的离线最优解具有(2 - 1/m)的竞争比，我们还证明了该竞争比是所有确定性算法的匹配下界；因此最优确定性公平竞争比恰好为2 - 1/m。在真实世界基准实例上的实验表明，Greedy-Balanced算法的实际表现优于其最坏情况保证，观测到的竞争比从未超过1.306。
英文摘要
We study interval scheduling from the perspective of fair allocation. There are $m$ identical machines and a set of intervals, each specified by a start time, an end time, and a nonnegative weight. A schedule assigns a subset of the intervals to the machines so that no two intervals on the same machine overlap, and the goal is to maximize the total weight of scheduled intervals. Viewing machines as agents and intervals as goods, we require the schedule to be envy-free up to one item (EF1), and we measure efficiency against the offline optimum without fairness. In the offline setting, we give an algorithm that computes an EF1 schedule whose loss is at most a factor of $3/2$ in the unweighted regime, and we prove lower bounds of $\frac{3m-2}{2m-1}$, approaching $3/2$, in both the unweighted and the unit-length weighted regimes, so the price of fairness is $3/2$ in the limit. In the online setting, intervals arrive in nondecreasing order of start times; an arriving interval must be accepted or rejected, rejections are irrevocable, and an accepted interval may be revoked, and lost, at any time before it ends. For the unweighted regime we present Greedy-Balanced, a simple algorithm that maintains EF1 at every point in time and is $(2-\tfrac{1}{m})$-competitive against the offline optimum without fairness, and we prove a matching lower bound for every deterministic algorithm; the optimal deterministic fair competitive ratio is thus exactly $2-\tfrac{1}{m}$. Experiments on real-world benchmark instances show that Greedy-Balanced performs well beyond its worst-case guarantee, with an observed ratio never exceeding $1.306$.
