We settle the pseudo-polynomial complexity of the Demand Strip Packing (DSP) problem: Given a strip of fixed width and a set of items with widths and heights, the items must be placed inside the strip with the objective of minimizing the peak height. This problem has gained significant scientific interest due to its relevance in smart grids[Deppert et al.\ APPROX'21, G\'alvez et al.\ APPROX'21]. Smart Grids are a modern form of electrical grid that provide opportunities for optimization. They are forecast to impact the future of energy provision significantly. Algorithms running in pseudo-polynomial time lend themselves to these applications as considered time intervals, such as days, are small. Moreover, such algorithms can provide superior approximation guarantees over those running in polynomial time. Consequently, they evoke scientific interest in related problems. We prove that Demand Strip Packing is strongly NP-hard for approximation ratios below $5/4$. Through this proof, we provide novel insights into the relation of packing and scheduling problems. Using these insights, we show a series of frameworks that solve both Demand Strip Packing and Parallel Task Scheduling optimally when increasing the strip's width or number of machines. Such alterations to problems are known as resource augmentation. Applications are found when penalty costs are prohibitively large. Finally, we provide a pseudo-polynomial time approximation algorithm for DSP with an approximation ratio of $(5/4+\varepsilon)$, which is nearly optimal assuming $P\neq NP$. The construction of this algorithm provides several insights into the structure of DSP solutions and uses novel techniques to restructure optimal solutions.

翻译：我们解决了需求条带包装(DSP)问题的伪多项式复杂度：给定一个固定宽度的条带和一组具有宽度和高度的物品，这些物品必须放置在条带内，目标是最小化峰值高度。由于该问题在智能电网中的相关性，它引起了科学界的广泛关注[Deppert等人, APPROX'21; Gálvez等人, APPROX'21]。智能电网是一种现代化的电网形式，为优化提供了机会，预计将显著影响未来能源供应的格局。运行在伪多项式时间内的算法适用于这些应用，因为所考虑的时间间隔（例如天数）较短。此外，这类算法相比多项式时间算法能提供更优的近似保证，因此引发了相关问题的科学兴趣。我们证明，当近似比低于$5/4$时，需求条带包装问题是强NP困难的。通过这一证明，我们提供了关于包装和调度问题之间关系的新见解。利用这些见解，我们展示了一系列框架，这些框架在增加条带宽度或机器数量时，能最优地解决需求条带包装问题和并行任务调度问题。这种对问题的修改被称为资源增强。当惩罚成本过高时，此类应用具有实际意义。最后，我们为DSP问题提供了一个近似比为$(5/4+\varepsilon)$的伪多项式时间近似算法，该算法在假设$P\neq NP$的情况下几乎是紧最优的。该算法的构建为DSP解的结构提供了若干见解，并采用了重新构建最优解的新技术。