ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines

TL;DR

This paper introduces an ETH-tight fixed-parameter tractable algorithm for minimizing makespan on uniform machines, improving previous bounds and employing innovative integer programming and modular arithmetic techniques.

Contribution

It presents the first ETH-tight FPT algorithm for makespan minimization on uniform machines, resolving an open question and enhancing computational efficiency.

Findings

Achieves an algorithm with runtime $p_{max}^{O(d)} n^{O(1)}$

Improves previous algorithms with quadratic dependence on $d$

Provides efficient implementation for high-multiplicity instances

Abstract

Given $n$ jobs with processing times $p_1,\dotsc,p_n\in\mathbb N$ and $m\le n$ machines with speeds $s_1,\dotsc,s_m\in\mathbb N$ 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 $d\le p_{\max}$ 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.