2025/01/08 by Rohwedder, Lars
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2501.04859
Given n jobs with processing times p1,\dotsc,pn∈\mathbb N and m≤ n machines with speeds s1,\dotsc,sm∈\mathbb N our goal is to allocate the jobs to machines minimizing the makespan. We present an algorithm that solves the problem in time pmaxO(d) nO(1), where pmax is the maximum processing time and d≤ pmax is the number of distinct processing times. This is essentially the best possible due to a lower bound based on the exponential time hypothesis (ETH). Our result improves over prior works that had a quadratic term in d in the exponent and answers an open question by Koutecký and Zink. The algorithm is based on integer programming techniques combined with novel ideas based on modular arithmetic. They can also be implemented efficiently for the more compact high-multiplicity instance encoding.