非对称加权提前-延误调度：具有非限制性公共交期的问题
Asymmetric Weighted Earliness-Tardiness: Scheduling with a Nonrestrictive Common Due Date
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中文总结 AI 辅助
针对单机非对称加权提前-延误调度问题，在非限制性公共交期下，证明了强NP完全性，并给出近似算法、伪多项式动态规划和FPTAS，揭示比率序交互的可控性。
中文摘要 AI 辅助
单机非对称加权提前-延误（AWET）调度问题研究如何围绕一个共同的同步日期对作业进行排序，此时提前或延迟完成会产生不相关的、依赖于作业的惩罚。在边界非限制性日期 $d=\sum_jp_j$ 处，一个紧凑的V形调度将连续时间问题简化为对非空提前集合的二次选择。我们针对该模型建立了四个互补的结果。首先，正整数问题通过从受限精确覆盖3-集合问题的单进制多项式归约，被证明是强NP完全的。其次，无限制AWET问题允许一个基于锚定半定松弛和确定性边际阈值的多项式时间 $(3+2\sqrt2+\varepsilon)$ 近似算法。第三，当提前和延误的比率序为严格反转时，该问题是弱NP完全的，但存在一个精确的双资源伪多项式动态规划。第四，对于比率排列可分的固定总细化，一个精确的分隔树递推和协调的几何修剪产生了一个FPTAS。证明使用了同一规范目标的不同表现形式：尺度分离的前缀惩罚、半正定最小核协方差、一个占优的已完成负载平方，以及一个有界的四坐标分解接口。综合来看，这些结果表明，决定性问题不仅在于两个比率序是否一致，还在于它们的相互作用能否被全局证书控制或压缩到有界的构造性接口中。
英文摘要
Single-machine asymmetric weighted earliness--tardiness (AWET) scheduling asks how to sequence jobs around a common synchronization date when early and late completion incur unrelated job-dependent penalties. At the boundary nonrestrictive date $d=\sum_jp_j$, a compact V-shaped schedule reduces the continuous-time problem to a quadratic choice of a nonempty early set. We establish four complementary results for this model. First, the positive-integer problem is strongly NP-complete by a unary-polynomial reduction from Restricted Exact Cover by 3-Sets. Second, unrestricted AWET admits a polynomial-time $(3+2\sqrt2+\varepsilon)$-approximation based on an anchored semidefinite relaxation and deterministic marginal thresholding. Third, when the earliness and tardiness ratio orders are strict reversals, the problem is weakly NP-complete but has an exact two-resource pseudopolynomial dynamic program. Fourth, for fixed total refinements whose ratio permutation is separable, an exact separating-tree recurrence and coordinated geometric trimming yield an FPTAS. The proofs use different manifestations of the same canonical objective: scale-separated prefix penalties, positive-semidefinite minimum-kernel covariance, a dominant completed load square, and a bounded four-coordinate decomposition interface. Together, the results show that the decisive issue is not merely whether the two ratio orders agree, but whether their interaction can be controlled by a global certificate or compressed into a bounded constructive interface.
发表机构
- The Ohio State University（俄亥俄州立大学）
- University of Graz, Austria（格拉茨大学）
- Hong Kong Polytechnic University, Hong Kong（香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。
