Scheduling Unrelated Machines of Few Different Types

TL;DR

This paper develops polynomial time approximation schemes for scheduling jobs on unrelated machines with a limited number of machine types, improving upon previous results for specific machine configurations.

Contribution

It introduces PTASs for minimizing makespan and L_p-norm in a setting with a fixed number of machine types, generalizing prior work.

Findings

PTASs for multidimensional jobs with fixed dimensions

PTASs for minimizing L_p-norm

Generalizes existing PTASs for related machine models

Abstract

A very well-known machine model in scheduling allows the machines to be unrelated, modelling jobs that might have different characteristics on each machine. Due to its generality, many optimization problems of this form are very difficult to tackle and typically APX-hard. However, in many applications the number of different types of machines, such as processor cores, GPUs, etc. is very limited. In this paper, we address this point and study the assignment of jobs to unrelated machines in the case that each machine belongs to one of a fixed number of types and the machines of each type are identical. We present polynomial time approximation schemes (PTASs) for minimizing the makespan for multidimensional jobs with a fixed number of dimensions and for minimizing the L_p-norm. In particular, our results subsume and generalize the existing PTASs for a constant number of unrelated machines and for an arbitrary number of identical machines for these problems. We employ a number of techniques which go beyond the previously known results, including a new counting argument and a method for making the concept of sparse extreme point solutions usable for a convex program.