The 2-valued case of makespan minimization with assignment constraints

TL;DR

This paper addresses a special case of makespan minimization with assignment constraints where jobs are either big or small, providing a polynomial-time approximation algorithm with a factor of 1.883 and improvements for limited machine assignments.

Contribution

It introduces a polynomial-time approximation algorithm for 2-valued makespan minimization with assignment constraints, achieving a 1.883 approximation ratio and enhancements for jobs with limited machine options.

Findings

Approximation ratio of 1.883 for the problem

Polynomial-time algorithm developed for the 2-valued case

Improved results when jobs are on at most two machines

Abstract

We consider the following special case of minimizing makespan. A set of jobs $J$ and a set of machines $M$ are given. Each job $j \in J$ can be scheduled on a machine from a subset $M_j$ of $M$. The processing time of $j$ is the same on all machines in $M_j.$ The jobs are of two sizes, namely $b$ (big) and $s$ (small). We present a polynomial-time algorithm that approximates the value of the optimal makespan within a factor of 1.883 and some further improvements when every job can be scheduled on at most two machines.