Breaking the Temporal Complexity Barrier: Bucket Calculus for Parallel Machine Scheduling
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918318147895296 |
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| author | Mohammad, Noor Islam S. |
| author_facet | Mohammad, Noor Islam S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01356 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Breaking the Temporal Complexity Barrier: Bucket Calculus for Parallel Machine Scheduling Mohammad, Noor Islam S. Computational Complexity 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. |
| title | Breaking the Temporal Complexity Barrier: Bucket Calculus for Parallel Machine Scheduling |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2602.01356 |