AI 中文总结
本文针对两个经典NP难多机调度问题，将其求解时间分别改进为${O}(2.755^n)$和${O}^\star(2^n)$，还在ARC假设下优化了装箱问题的求解时间。
AI 中文摘要
在多台相同机器上最小化加权完成时间（$P \mid \mid \Sigma w_j C_j$）和加权延迟作业数（$P \mid \mid \Sigma w_j U_j$）是两个经典的NP难调度问题。Lenté等人（2014）证明这两个问题均可在${O}^\star(3^n)$时间内求解。本文将这两个问题的求解时间分别改进为${O}(2.755^n)$和${O}^\star(2^n)$。针对$P \mid \mid \Sigma w_j C_j$的算法采用了中途相遇（meet-in-the-middle）范式和一种能高效回答线性规划查询的数据结构；此外，当机器数量不超过6时，本文证明$P \mid \mid \Sigma w_j C_j$的运行时间可进一步优化。这两个调度问题都是经典的装箱问题（Bin Packing）的推广，装箱问题可在${O}^\star(2^n)$时间内求解，改进该运行时间是一个重要的开放问题。本文证明，在渐近秩猜想（Asymptotic Rank Conjecture, ARC）成立的假设下，装箱问题可在${O}((2-\varepsilon)^n)$时间内求解（其中$\varepsilon >0$）。本文的算法包含两个核心部分：一是Nederlof等人[SICOMP'23]提出的当箱子数量为固定常数时装箱问题的${O}((2-\varepsilon)^n)$时间算法，二是Björklund等人[SODA'25]提出的在ARC假设下3路划分问题（3-way Partitioning）特殊实例的${O}((2-\varepsilon)^n)$时间算法。
英文摘要
Minimizing the weighted completion times ($P \mid \mid Σw_j C_j$) and weighted number of tardy jobs ($P \mid \mid Σw_j U_j$) on multiple identical machines are two classical NP-hard scheduling problems. As shown by Lenté et al. (2014), both problems can be solved in time ${O}^{\star}(3^n)$. In this paper, we improve these bounds to ${O}(2.755^n)$ and ${O}^{\star}(2^n)$, respectively. Our algorithm for $P \mid \mid Σw_j C_j$ exploits the meet-in-the-middle paradigm and an efficient data structure answering linear programming queries. Additionally, when the number of machines is at most $6$, we show that the running time for $P \mid \mid Σw_j C_j$ can further be improved. Both scheduling problems are generalizations of the classical Bin Packing problem, which can be solved in ${O}^{\star}(2^n)$ time. Improving this running time is an important open question. We show that, when assuming the Asymptotic Rank Conjecture (ARC), Bin Packing can be solved in time ${O}((2-\varepsilon)^n)$ for some $\varepsilon >0$. Our algorithm makes use of two main ingredients: the recent ${O}((2-\varepsilon)^n)$-time algorithm of Nederlof et al. [SICOMP'23] for Bin Packing when the number of bins is a fixed constant, and the ${O}((2-\varepsilon)^n)$-time algorithm of Björklund et al. [SODA'25] for special instances of the $3$-way Partitioning problem when assuming ARC.
DOI：10.4230/LIPIcs.ESA.2026.53
