AI 中文总结
针对少机器离线临时任务分配问题，提出EPTAS，运行时间$f(r,1/\varepsilon)\cdot N^{O(1)}$，并证明强NP难及无FPTAS，关键用预置加权区间和动态规划。
AI 中文摘要
在离线临时任务分配问题中，每个作业有一个时间区间和一个正权重，并被分配给 $r$ 台相同机器中的一台，在其整个区间内运行；目标是最小化峰值负载，即一台机器上同时活跃作业的总权重的最大值。对于每个固定的 $r$，我们给出一个高效多项式时间近似方案（EPTAS），其运行时间为 $f(r,1/\varepsilon)\cdot N^{O(1)}$，其中 $N$ 是输入长度。之前的近似方案由 Azar、Regev、Sgall 和 Woeginger（2002）提出，在 $n$ 个作业上运行时间为 $n^{O(r^3\log r/\varepsilon^3)}$。我们还证明了对于每个固定的 $r\ge 2$，该问题是强 NP 困难的，因此除非 P=NP，否则不存在 FPTAS；并且在指数时间假设下，不存在确定性 $(1+\varepsilon)$-近似算法能在 $2^{o(1/\varepsilon)}\cdot N^{O(1)}$ 时间内运行。关键思想是用少量权重区间来限制每组同时活跃作业的负载，这些区间的位置是预先固定的；然后，时间线上的动态规划只需记录每台机器承载哪些区间。
英文摘要
We study vector scheduling in which each job is active during a fixed time interval. A job uses several resources and stays on one machine for its entire interval; its resource requirements may depend on the machine. The objective is to minimize the largest resource load over all machines and times. For $r$ machines and $d$ resources, we give a deterministic $(1+\varepsilon)$-approximation in $f(r,d,1/\varepsilon)N^{O(1)}$ time, where $N$ is the binary input length. This gives an efficient polynomial-time approximation scheme for fixed $r$ and $d$, extending approximation schemes for scalar temporary tasks assignment. The algorithm merges jobs into blocks whose time intervals are fixed before any machine is chosen, and assigns the blocks by dynamic programming over a balanced recursive split of the time line. We also prove strong NP-hardness and an exponential lower bound in $1/\varepsilon$ under the Exponential Time Hypothesis, already for two identical machines and one resource.
Comments18 pages, 4 figures. v2: extended to several resources and unrelated machines, with a new title and simplified proofs
