Improved Approximations for Multiprocessor Scheduling Under Uncertainty

TL;DR

This paper introduces improved approximation algorithms for multiprocessor scheduling under uncertainty, reducing the approximation ratio for various special cases and advancing the state of the art in this NP-hard problem.

Contribution

The paper presents asymptotically better approximation algorithms for multiprocessor scheduling under uncertainty, notably improving ratios for independent jobs and chain-structured precedence constraints.

Findings

Achieved an O(loglog min(m,n))-approximation for independent jobs.

Developed an O(log(n+m) loglog min(m,n))-approximation for chain precedence constraints.

Provided algorithms that extend to precedence trees, improving previous bounds.

Abstract

This paper presents improved approximation algorithms for the problem of multiprocessor scheduling under uncertainty, or SUU, in which the execution of each job may fail probabilistically. This problem is motivated by the increasing use of distributed computing to handle large, computationally intensive tasks. In the SUU problem we are given n unit-length jobs and m machines, a directed acyclic graph G of precedence constraints among jobs, and unrelated failure probabilities q_{ij} for each job j when executed on machine i for a single timestep. Our goal is to find a schedule that minimizes the expected makespan, which is the expected time at which all jobs complete. Lin and Rajaraman gave the first approximations for this NP-hard problem for the special cases of independent jobs, precedence constraints forming disjoint chains, and precedence constraints forming trees. In this paper, we present asymptotically better approximation algorithms. In particular, we give an O(loglog min(m,n))-approximation for independent jobs (improving on the previously best O(log n)-approximation). We also give an O(log(n+m) loglog min(m,n))-approximation algorithm for precedence constraints that form disjoint chains (improving on the previously best O(log(n)log(m)log(n+m)/loglog(n+m))-approximation by a (log n/loglog n)^2 factor when n = poly(m). Our algorithm for precedence constraints forming chains can also be used as a component for precedence constraints forming trees, yielding a similar improvement over the previously best algorithms for trees.