发表机构
University of Maryland; The Ohio State University（马里兰大学; 俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明带释放时间和相同处理时间的加权延误问题为强NP完全，并给出相位网格分配算法，达到3/2-1/(2N)的紧近似比，解决了自2010年以来的开放问题。
AI 中文摘要
我们研究单机上带释放时间、截止日期、正作业权重和共同处理时间的非抢占式调度问题。目标是最小化总加权延误。尽管密切相关且处理时间相等的相关问题存在多项式时间算法，但该问题的复杂性自2010年以来在文献中一直悬而未决。我们证明其判定版本是强NP完全的，即使每个作业在释放时立即处理也能满足其截止日期。该归约来自无权MAX-CUT问题，使用二次数量级的作业，其数值数据具有多项式有界性。其主要成分是一个构造性规范化定理，该定理将每个足够廉价的可行调度转换为每个图顶点的二元选择；规范化后，总加权延误等于一个常数减去缩放后的割值。我们还针对平移目标Φ=F+p∑_j w_j给出了一个确定性的多项式时间相位网格分配算法，其中F是总加权延误。该算法最多枚举N个模p的释放时间余数，为每个余数求解一个最小成本分配问题，并返回最佳的相位网格调度。它运行在O(N^5)次算术运算内，并为此算法达到紧比率3/2-1/(2N)。由于附加项p∑_j w_j与作业的调度方式无关，平移目标和原始目标具有完全相同的调度方案。然而，近似保证适用于平移目标；对于原始目标，分析提供了加性界。因此，本文既解决了长期存在的复杂性难题，又为相位网格分配算法提供了互补的最坏情况保证。
英文摘要
We study nonpreemptive scheduling on a single machine with release dates, due dates, positive job weights, and a common processing time. The objective is to minimize total weighted tardiness. Although closely related equal-processing-time problems admit polynomial-time algorithms, the complexity of this problem has remained open in the literature since 2010. We prove that its decision version is strongly NP-complete, even when every job can meet its due date if processed immediately upon release. The reduction is from unweighted MAX-CUT and uses a quadratic number of jobs with polynomially bounded numerical data. Its main ingredient is a constructive normalization theorem that converts every sufficiently inexpensive feasible schedule into a binary choice for each graph vertex; after normalization, total weighted tardiness equals a constant minus a scaled cut value. We also give a deterministic polynomial-time phase-grid assignment algorithm for the shifted objective $Φ=F+p\sum_jw_j$, where $F$ is total weighted tardiness. The algorithm enumerates at most $N$ release-date residues modulo $p$, solves one minimum-cost assignment problem for each residue, and returns the best phase-grid schedule. It runs in $O(N^5)$ arithmetic operations and achieves the tight ratio $3/2-1/(2N)$ for this algorithm. Because the added term $p\sum_jw_j$ is independent of how the jobs are scheduled, the shifted and original objectives have exactly the same optimal schedules. However, the approximation guarantee applies to the shifted objective; for the original objective, the analysis provides an additive bound. Thus, the paper both resolves the long-standing complexity question and provides a complementary worst-case guarantee for the phase-grid assignment algorithm.
Comments38 pages, 2 figures, 5 tables
