ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines

2025-01-08

AI Summary

This paper studies the problem of minimizing makespan on uniform parallel machines: scheduling $n$ jobs with distinct processing times onto $m$ machines having different speeds. For instances where the number of distinct processing times is $d$ and the maximum processing time is $p_{max}$, we present the first ETH-tight fixed-parameter tractable (FPT) algorithm, running in time $p_{max}^{O(d)} cdot n^{O(1)}$. This breaks the previous exponential barrier of $p_{max}^{O(d^2)}$ and resolves the open problem posed by Koutecký and Zink. Our approach formulates the problem as an integer program and introduces a novel modular arithmetic technique to enable compact high-multiplicity encoding and efficient implementation. The algorithm’s runtime matches the theoretical lower bound implied by the Exponential Time Hypothesis (ETH), establishing optimal asymptotic dependence on $p_{max}$ and $d$. This advancement significantly improves the scalability and practical efficiency for solving large-scale heterogeneous scheduling instances.

Technology Category

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

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: Fairness and ethical considerations in crowd work and in human-in-the-loop AI systems

Abstract

Given $n$ jobs with processing times $p_1,dotsc,p_ninmathbb N$ and $mle n$ machines with speeds $s_1,dotsc,s_minmathbb N$ our goal is to allocate the jobs to machines minimizing the makespan. We present an algorithm that solves the problem in time $p_{max}^{O(d)} n^{O(1)}$, where $p_{max}$ is the maximum processing time and $dle p_{max}$ 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 Kouteck'y 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.

Problem

• Task Scheduling
• Optimization
• Machine Speed Variability

Innovation

• ETH Algorithm
• Optimal Task Allocation
• Computational Efficiency