Scheduling with Outliers

Abstract

In classical scheduling problems, we are given jobs and machines, and have to schedule all the jobs to minimize some objective function. What if each job has a specified profit, and we are no longer required to process all jobs? Instead, we can schedule any subset of jobs whose total profit is at least a (hard) target profit requirement, while still trying to approximately minimize the objective function. We refer to this class of problems as scheduling with outliers. This model was initiated by Charikar and Khuller (SODA ’06) for minimum max-response time in broadcast scheduling. In this paper, we consider three other well-studied scheduling objectives: the generalized assignment problem, average weighted completion time, and average flow time, for which LP-based approximation algorithms are provided. Our main results are: For the minimum average flow time problem on identical machines, we give an LP-based logarithmic approximation algorithm for the unit profits case, and complement this result by presenting a matching integrality gap. For the average weighted completion time problem on unrelated machines, we give a constant-factor approximation. The algorithm is based on randomized rounding of the time-indexed LP relaxation strengthened by knapsack-cover inequalities. For the generalized assignment problem with outliers, we outline a simple reduction to GAP without outliers to obtain an algorithm whose makespan is within 3 times the optimum makespan, and whose cost is at most (1 + ε) times the optimal cost.