Techniques for Scheduling with Rejection

We consider the general problem of scheduling a set of jobs where we may choose not to schedule certain jobs, and thereby incur a penalty for each rejected job. More specifically, we focus on choosing a set of jobs to reject and constructing a schedule for the remaining jobs so as to optimize the sum of the weighted completion times of the jobs scheduled plus the sum of the penalties of the jobs rejected. We give several techniques for designing scheduling algorithms under this criterion. Many of these techniques show how to reduce a problem with rejection to a (potentially more complex) scheduling problem without rejection. Some of the reductions are based on general properties of certain kinds of linear-programming relaxations of optimization problems, and therefore are applicable to problems outside of scheduling; we demonstrate this by giving an approximation algorithm for a variant of the facility-location problem. In the last section of the paper we consider a different notion of rejection in the context of scheduling: scheduling jobs with due dates so as to maximize the number of jobs that complete by their due dates, or equivalently to minimize the number of jobs that do not complete by their due date and that thus can be considered “rejected.” We investigate the approximability of a simple version of this problem, giving approximation algorithms and characterizing integrality gaps of a class of linear-programming relaxations. ⋆ [email protected] of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139. Research supported by NSF Contract MIP-9612632. ⋆⋆ {karger, sudipta}@theory.lcs.mit.edu. Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139. Research supported in part by ARPA contract N00014-95-1-1246 and NSF contract CCR-9624239, as well as grants from the Alfred P. Sloane and David and Lucille Packard foundations. ⋆ ⋆ ⋆ [email protected] of Computer Science, Dartmouth College, Hanover, NH 03755-3510. Research partially supported by NSF Award CCR-9308701 and NSF Career Award CCR-9624828. † [email protected]. Department of Computer Science, Polytechnic University, Brooklyn, NY, 11201. Research partially supported by NSF Grant CCR-9626831. ‡ [email protected]. Department of Computer Science, Polytechnic University, Brooklyn, NY, 11201. Research partially supported by NSF Grant CCR-9626831 and a grant from the New York State Science and Technology Foundation, through its Center for Advanced Technology in Telecommunications.