Breaking the Temporal Complexity Barrier: Bucket Calculus for Parallel Machine Scheduling

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.

💡 Research Summary

The paper tackles the classic parallel machine scheduling problem P m | r_j | C_max, focusing on the strongly NP‑hard case with two machines (P2|r_j|C_max). Traditional exact approaches rely on time‑indexed mixed‑integer linear programming (MILP) formulations that discretize the planning horizon at unit‑time granularity, leading to a variable count on the order of O(T·|J|·|M|) and computational complexity O(Tⁿ). This makes them infeasible for industrial‑scale instances where the horizon T can exceed 10⁴.

The authors introduce “bucket calculus,” a novel mathematical framework that compresses the temporal dimension into a set of coarser “buckets.” The horizon is partitioned using a granularity Δ equal to the smallest processing time among jobs, yielding B = ⌊T/Δ⌋ + 1 buckets, where B ≪ T. By separating exact combinatorial decisions (job‑to‑machine assignment and sequencing) from approximate temporal positioning, they formulate a bucket‑indexed MILP that retains the essential structure of the original problem while dramatically reducing the number of binary variables (by about 94.4 %).

Key theoretical contributions include: (1) Partial discretization theory, which proves that for any feasible schedule in the original space there exists a corresponding schedule in the compressed space whose makespan deviates by at most a pre‑specified ε; (2) Fractional bucket calculus operators (difference operator ∇ and bucket transform B