Tighter Bounds for Makespan Minimization on Unrelated Machines

TL;DR

This paper introduces a new polynomial-time algorithm that provides tighter approximation bounds for makespan minimization on unrelated machines, especially for instances with high feasibility, improving upon the classic 2-approximation.

Contribution

It develops a novel algorithm that leverages the feasibility factor to achieve better bounds for a broad class of instances, advancing the understanding of scheduling approximations.

Findings

For instances with high feasibility factor, the algorithm finds schedules with makespan less than 2T.

The approach either proves the non-existence of certain schedules or finds improved schedules.

For restricted job-machine processing times, the algorithm surpasses previous approximation ratios.

Abstract

We consider the problem of scheduling $n$ jobs to minimize the makespan on $m$ unrelated machines, where job $j$ requires time $p_{ij}$ if processed on machine $i$. A classic algorithm of Lenstra et al. yields the best known approximation ratio of $2$ for the problem. Improving this bound has been a prominent open problem for over two decades. In this paper we obtain a tighter bound for a wide subclass of instances which can be identified efficiently. Specifically, we define the feasibility factor of a given instance as the minimum fraction of machines on which each job can be processed. We show that there is a polynomial-time algorithm that, given values $L$ and $T$, and an instance having a sufficiently large feasibility factor $h \in (0,1]$, either proves that no schedule of mean machine completion time $L$ and makespan $T$ exists, or else finds a schedule of makespan at most $T + L/h < 2T$. For the restricted version of the problem, where for each job $j$ and machine $i$, $p_{ij} \in \{p_j, \infty\}$, we show that a simpler algorithm yields a better bound, thus improving for highly feasible instances the best known ratio of $33/17 + \epsilon$, for any fixed $\epsilon >0$, due to Svensson.