Faster Exponential Algorithms for Multi-Machine Scheduling Problems

2026-08-12

AI Summary

This work addresses the NP-hard scheduling problems of minimizing the total weighted completion time (P||ΣwⱼCⱼ) and the total weighted number of tardy jobs (P||ΣwⱼUⱼ) on identical parallel machines, as well as the related Bin Packing problem, all of which suffer from exponential-time bottlenecks. By integrating a meet-in-the-middle strategy, an efficient linear programming query data structure, and the Asymptotic Rank Conjecture (ARC), the authors achieve the first improvements in worst-case time complexity: reducing P||ΣwⱼCⱼ to O(2.755ⁿ) and P||ΣwⱼUⱼ to O*(2ⁿ), with further refinements when the number of machines is at most six. Moreover, under the ARC, they conditionally obtain an exponential speedup for Bin Packing, yielding a running time of O((2−ε)ⁿ). These results represent a significant theoretical advance over the previous best-known bound of O*(3ⁿ).

Abstract

Minimizing the weighted completion times ($P \mid \mid Σw_j C_j$) and weighted number of tardy jobs ($P \mid \mid Σw_j U_j$) on multiple identical machines are two classical NP-hard scheduling problems. As shown by Lenté et al. (2014), both problems can be solved in time ${O}^{\star}(3^n)$. In this paper, we improve these bounds to ${O}(2.755^n)$ and ${O}^{\star}(2^n)$, respectively. Our algorithm for $P \mid \mid Σw_j C_j$ exploits the meet-in-the-middle paradigm and an efficient data structure answering linear programming queries. Additionally, when the number of machines is at most $6$, we show that the running time for $P \mid \mid Σw_j C_j$ can further be improved. Both scheduling problems are generalizations of the classical Bin Packing problem, which can be solved in ${O}^{\star}(2^n)$ time. Improving this running time is an important open question. We show that, when assuming the Asymptotic Rank Conjecture (ARC), Bin Packing can be solved in time ${O}((2-\varepsilon)^n)$ for some $\varepsilon >0$. Our algorithm makes use of two main ingredients: the recent ${O}((2-\varepsilon)^n)$-time algorithm of Nederlof et al. [SICOMP'23] for Bin Packing when the number of bins is a fixed constant, and the ${O}((2-\varepsilon)^n)$-time algorithm of Björklund et al. [SODA'25] for special instances of the $3$-way Partitioning problem when assuming ARC.

Problem

• multi-machine scheduling
• weighted completion time
• weighted tardy jobs
• Bin Packing
• exponential algorithms

Innovation

• exponential-time algorithms
• scheduling
• meet-in-the-middle
• Bin Packing
• Asymptotic Rank Conjecture