ETH-Tight FPT Algorithm for Makespan Minimization on Uniform Machines

TL;DR

The paper studies Makespan Minimization on Uniform Machines, a strongly NP-hard scheduling problem. It delivers an ETH-tight fixed-parameter tractable algorithm with running time $p_{ ext{max}}^{O(d)} n^{O(1)}$, where $d$ is the number of distinct processing times, by reducing to a novel Modulo Integer Programming formulation and a Multiway Partitioning subproblem solved via a Multi-Choice Integer Programming framework that leverages the Steinitz Lemma. The approach unifies modular feasibility with a constructive recovery of a full assignment, and extends efficiently to high-multiplicity encodings, improving the exponent from $O(d^{2})$ to $O(d)$ and answering an open question of Koutecký and Zink. This advances exact IP techniques for scheduling and highlights new ideas for leveraging modular constraints, configuration-based IPs, and proximity arguments in ETH-tight FPT algorithms.

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 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.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Lemma 6
• Proposition 7: Steinitz Lemma sevast1978approximate, see also eisenbrand2019proximity