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

TL;DR

This paper presents bucket calculus, a new framework that significantly reduces the complexity of parallel machine scheduling optimization, enabling near-optimal solutions for large-scale NP-hard problems through adaptive temporal discretization.

Contribution

Introduction of bucket calculus and adaptive temporal discretization that drastically reduce complexity and decision variables in scheduling optimization, with theoretical and empirical validation.

Findings

Achieved exponential complexity reduction from O(T^n) to O(B^n)

Reduced decision variables by 94.4% in MILP formulation

Demonstrated near-optimal solutions with less than 5.1% gap on large instances

Abstract

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 ($\sigma/\mu = 0.006$), 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.