This paper introduces bucket calculus, a novel mathematical framework that fundamentally transforms the computational complexity landscape of parallel machine scheduling optimization. We address the strongly NP-hard problem $P2|r_j|C_{\max}$ through an innovative adaptive temporal discretization methodology that achieves exponential complexity reduction from $O(T^n)$ to $O(B^n)$ where $B \ll T$, while maintaining near-optimal solution quality. Our bucket-indexed mixed-integer linear programming (MILP) formulation exploits dimensional complexity heterogeneity through precision-aware discretization, reducing decision variables by 94.4\% and achieving a theoretical speedup factor $2.75 \times 10^{37}$ for 20-job instances. Theoretical contributions include partial discretization theory, fractional bucket calculus operators, and quantum-inspired mechanisms for temporal constraint modeling. Empirical validation on instances with 20--400 jobs demonstrates 97.6\% resource utilization, near-perfect load balancing ($σ/μ= 0.006$), and sustained performance across problem scales with optimality gaps below 5.1\%. This work represents a paradigm shift from fine-grained temporal discretization to multi-resolution precision allocation, bridging the fundamental gap between exact optimization and computational tractability for industrial-scale NP-hard scheduling problems.

翻译：本文提出桶演算这一新颖的数学框架，从根本上改变了并行机调度优化的计算复杂性图景。我们通过创新的自适应时间离散化方法处理强NP难问题$P2|r_j|C_{\max}$，在保持接近最优解质量的同时，将指数级复杂性从$O(T^n)$降至$O(B^n)$，其中$B \ll T$。我们基于桶索引的混合整数线性规划（MILP）公式通过精度感知离散化利用维度复杂性异质性，将决策变量减少94.4%，并在20个作业实例上实现$2.75 \times 10^{37}$的理论加速因子。理论贡献包括部分离散化理论、分数桶演算算子以及用于时间约束建模的量子启发机制。在20至400个作业实例上的实证验证表明，该方法实现了97.6%的资源利用率、接近完美的负载平衡（$σ/μ= 0.006$），并在问题规模扩展时保持稳定性能，最优性差距低于5.1%。这项工作代表了从细粒度时间离散化到多分辨率精度分配的范式转变，为工业规模NP难调度问题的精确优化与计算可处理性之间架起了根本性桥梁。