Fast algorithms for parametric scheduling come from extensions to parametric maximum flow

Abstract

Chen [3] develops an attractive variant of the classical problem of preemptively scheduling independent jobs with release dates and due dates. Chen suggests that in practice one can often pay to reduce the processing requirement of a job. This leads to two parametric max flow problems. Serafini [14] considers scheduling independent jobs with due dates on multiple machines, where jobs can be split among machines so that pieces of a single job can execute in parallel. Minimizing the maximum tardiness again gives a parametric max flow problem. A third problem of this type is deciding how many more games a baseball team can lose partway through a season without being eliminated from finishing first. A fourth such problem is an extended selection problem of Brumelle, Granot, and Liu [2], where we want to discount the costs of "tree-structured" tools as little as possible to be able to process all jobs at a profit. It is tempting to try to solve these problems with the parametric push-relabel max flow methods of Gallo, Grigoriadis and Tarjan (GGT) [6]. However, all of these applications appear to violate the conditions necessary to apply GGT. We extend GGT in three ways which allow it to be applied to all four of the above applications. Our extensions to GGT yield faster algorithms for all these applications.