On Minimizing Generalized Makespan on Unrelated Machines

TL;DR

This paper investigates the complexity of minimizing the generalized makespan on unrelated machines, proving it is hard to approximate within a logarithmic factor and providing bounds on the integrality gap of related LP relaxations.

Contribution

It establishes the hardness of approximating the GMP for general norms and presents new bounds on the integrality gap of the configuration LP.

Findings

GMP is $ ext{Omega}( ext{log}^{ ext{delta}} n)$ hard to approximate.

Configuration LP has an $ ext{Omega}( ext{log}^{1/2} n)$ integrality gap.

Previous approximation algorithms are limited by these hardness results.

Abstract

We consider the Generalized Makespan Problem (GMP) on unrelated machines, where we are given $n$ jobs and $m$ machines and each job $j$ has arbitrary processing time $p_{ij}$ on machine $i$. Additionally, there is a general symmetric monotone norm $\psi_i$ for each machine $i$, that determines the load on machine $i$ as a function of the sizes of jobs assigned to it. The goal is to assign the jobs to minimize the maximum machine load. Recently, Deng, Li, and Rabani (SODA'22) gave a $3$ approximation for GMP when the $\psi_i$ are top-$k$ norms, and they ask the question whether an $O(1)$ approximation exists for general norms $\psi$? We answer this negatively and show that, under natural complexity assumptions, there is some fixed constant $\delta>0$, such that GMP is $\Omega(\log^{\delta} n)$ hard to approximate. We also give an $\Omega(\log^{1/2} n)$ integrality gap for the natural configuration LP.