在活跃时间预算约束下最大化加权吞吐量的调度问题
Scheduling to Maximize Weighted Throughput with an Active-Time Budget
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- Technical University of Munich（慕尼黑工业大学）
- TUM School of Computation, Information and Technology（计算、信息与技术学院）
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中文总结 AI 辅助
该研究针对活跃时间调度问题，在活跃时间预算K约束下最大化已完成作业总权重，针对不同区间类型（一般、正则、层状）给出了难度分析及对应近似或精确算法。
中文摘要 AI 辅助
我们研究活跃时间调度问题，目标是最大化加权吞吐量。该问题设定为：n个作业J在整数释放时间到达，每个作业有整数处理时间和整数截止时间，作业可在整数时隙边界被抢占；调度将作业分配到时隙，同一时隙最多分配m个作业，若时隙安排了至少一个作业则称为活跃时隙。与传统的最小化活跃时隙数量（即调度所有作业）不同，我们考虑带有活跃时间预算K的更通用加权吞吐量变体：每个作业j∈J有权重w_j，目标是在使用不超过K个活跃时隙的条件下，最大化已完成作业的总权重，未完成部分的作业不计入目标。传统活跃时间最小化问题可转化为判断是否能在给定活跃时间预算内完成所有作业。我们给出了该问题的难度、近似算法和精确算法结果：对于并行度无界的一般区间，我们证明其为NP难问题，除非P=NP否则不存在完全多项式时间近似方案（FPTAS），并给出伪多项式时间的Ω(1/log K)近似算法；对于正则区间，我们证明了一个典型结构引理，并得到精确的(nK)^(O(m))时间算法；对于层状区间，我们给出精确的f(K,m)·n^(O(1))时间算法。
英文摘要
We study the active-time scheduling problem with weighted throughput maximization. In this setting, a set of $n$ jobs $J$ arrive at integer release times, each with an integer processing time and integer deadline. Jobs may be preempted at integer time slot boundaries. A schedule assigns jobs to time slots, with at most $m$ jobs assigned to the same time slot. A slot is called \emph{active} if at least one job is scheduled in it. Instead of scheduling all jobs to minimize the number of active time slots, we consider the more general variant of \emph{weighted throughput} with an active-time budget $K$, where each job $j\in J$ has a weight $w_j$. The objective is to maximize the total weight of \emph{completed} jobs using at most $K$ active time slots. This means that partially scheduled jobs do not count towards the objective. The classical active-time minimization problem is recovered by asking whether all jobs can be completed within a given active-time budget. We give hardness, approximation, and exact algorithmic results. For general intervals with unbounded parallelism, we prove NP-hardness, rule out an FPTAS unless $\mathrm{P}=\mathrm{NP}$, and give a pseudo-polynomial time $Ω(1/\log K)$-approximation. For proper intervals, we prove a canonical structural lemma and obtain an exact $(nK)^{O(m)}$-time algorithm. For laminar intervals, we give an exact $f(K,m)\cdot n^{O(1)}$-time algorithm.
