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

2026-02-01

AI Summary

This study addresses the strongly NP-hard parallel machine scheduling problem $P2|r_j|C_{\max}$, which is notoriously difficult to scale due to the exponential time complexity $O(T^n)$ of conventional approaches. To overcome this limitation, the authors propose a bucket calculus framework that adaptively discretizes the time horizon, reducing complexity to $O(B^n)$ with $B \ll T$. Key innovations include bucket-indexed MILP modeling, precision-aware partial discretization, fractional bucket operators, and a quantum-inspired temporal constraint mechanism enabling multi-resolution precision allocation. Experimental results demonstrate a 94.4% reduction in decision variables and a theoretical speedup of up to $2.75 \times 10^{37}$. On instances with 20–400 tasks, the method achieves 97.6% resource utilization, excellent load balancing (σ/μ = 0.006), and an optimality gap below 5.1%.

Technology Category

• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• parallel machine scheduling
• strongly NP-hard
• temporal complexity
• computational tractability
• scheduling optimization

Innovation

• bucket calculus
• adaptive temporal discretization
• mixed-integer linear programming
• complexity reduction
• quantum-inspired scheduling