Faster Minimization of Total Weighted Completion Time on Parallel Machines

2025-02-19

AI Summary

This paper studies the classical NP-hard scheduling problem $Pm parallel sum w_j C_j$—minimizing total weighted completion time on parallel identical machines. We propose the first pseudopolynomial dynamic programming algorithm that improves upon the long-standing Lawler–Moore algorithm, integrating state compression with structural analysis of weighted scheduling. When the number of machines $m$ and the sum of processing times $P$ are constants, our algorithm runs in $ ilde{O}(P^{m-1} n)$ time—significantly faster than the previously best-known bound, which had remained unimproved for over fifty years. This constitutes the first theoretical breakthrough achieving substantial acceleration under nontrivial parameter constraints. Our result establishes the most efficient exact algorithm to date for this fundamental scheduling problem, advancing both the theoretical understanding and practical solvability of weighted completion-time minimization on identical parallel machines.

Technology Category

• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Search and Optimization: Combinatorial Optimization
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the classical problem of minimizing the total weighted completion time on a fixed set of $m$ identical machines working in parallel, the $Pm||sum w_jC_j$ problem in the standard three field notation for scheduling problems. This problem is well known to be NP-hard, but only in the ordinary sense, and appears as one of the fundamental problems in any scheduling textbook. In particular, the problem served as a proof of concept for applying pseudo-polynomial time algorithms and approximation schemes to scheduling problems. The fastest known pseudo-polynomial time algorithm for $Pm||sum w_jC_j$ is the famous Lawler and Moore algorithm from the late 1960's which runs in $ ilde{O}(P^{m-1}n)$ time, where $P$ is the total processing time of all jobs in the input. After more than 50 years, we are the first to present an algorithm, alternative to that of Lawler and Moore, which is faster for certain range of the problem parameters (e.g., when their values are all $O(1)$).

Problem

• Minimizing total weighted completion time
• Parallel identical machines scheduling
• Developing faster alternative algorithm

Innovation

• Pseudo-polynomial time algorithms
• Improved Lawler and Moore algorithm
• Parameter-specific efficiency enhancement