Scheduling with Outliers

In classical scheduling problems, we are given jobs and machines, and have to schedule all the jobsto minimize some objective function. What if each job has a specified profit, and we are no longerrequired to process all jobs—we can schedule any subset of jobs whose total profit is at least a (hard)target profit requirement, while still approximately minimizing the objective function?We refer to this class of problems as scheduling with outliers. This model was initiated by Charikar andKhuller (SODA’06) on the minimum max-response time in broadcast scheduling. In this paper, weconsider three other well-studied scheduling objectives: the generalized assignment problem, averageweighted completion time, and average flow time, and provide LP-based approximation algorithmsfor them. Our main results are:• For the minimum average flow time problem on identical machines, we give a logarithmicapproximation algorithm for the case of unit profits based on rounding an LP relaxation; wealso show a matching integrality gap. While the LP relaxation has been used before, therounding algorithm is a delicate one.• 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 LPrelaxation strengthened by the knapsack-cover inequalities.• For the generalized assignment problem with outliers, we give a simple reduction to GAP with-out outliers to obtain an algorithm whose makespan is within 3 times the optimum makespan,and whose cost is at most (1 + ǫ) times the optimal cost. Computer Science Department, Carnegie Mellon University. Supported in part by NSF awards CCF-0448095 andCCF-0729022, and an Alfred P. Sloan Fellowship.Department of Computer Science & Engineering, Indian Institute of Technology, Hauz Khas, New Delhi, India -110016. Work partly done at MPI, Saarbrücken, Germany.Sloan School of Management, Massachusetts Institute of Technology. Supported in part by NSF awards CCF-0448095and CCF-0729022, and an Alfred P. Sloan Fellowship.