Approximation Algorithms for Multiprocessor Scheduling under Uncertainty

TL;DR

This paper develops polynomial-time approximation algorithms for multiprocessor scheduling under uncertainty, minimizing expected makespan in various dependency graph structures, addressing an NP-hard problem.

Contribution

It introduces new approximation algorithms for scheduling under uncertainty with different dependency graph cases, improving understanding of the problem's complexity.

Findings

O(log(n))-approximation for independent jobs

O(log(m)log(n)log(n+m)/loglog(n+m))-approximation for disjoint chains

O(log(m)log^2(n))-approximation for directed trees

Abstract

Motivated by applications in grid computing and project management, we study multiprocessor scheduling in scenarios where there is uncertainty in the successful execution of jobs when assigned to processors. We consider the problem of multiprocessor scheduling under uncertainty, in which we are given n unit-time jobs and m machines, a directed acyclic graph C giving the dependencies among the jobs, and for every job j and machine i, the probability p_{ij} of the successful completion of job j when scheduled on machine i in any given particular step. The goal of the problem is to find a schedule that minimizes the expected makespan, that is, the expected completion time of all the jobs. The problem of multiprocessor scheduling under uncertainty was introduced by Malewicz and was shown to be NP-hard even when all the jobs are independent. In this paper, we present polynomial-time approximation algorithms for the problem, for special cases of the dag C. We obtain an O(log(n))-approximation for the case of independent jobs, an O(log(m)log(n)log(n+m)/loglog(n+m))-approximation when C is a collection of disjoint chains, an O(log(m)log^2(n))-approximation when C is a collection of directed out- or in-trees, and an O(log(m)log^2(n)log(n+m)/loglog(n+m))-approximation when C is a directed forest.