A Combination of Parallel Machine Scheduling and the Covering Problem

This paper studies a combination problem of parallel machine scheduling and the covering problem. We say a combinatorial optimization problem is a covering problem if it minimizes a linear objective function under some constraints and the variables are binary. The covering problem generalizes many classic combinatorial optimization problems, e.g. the shortest path problem, the vertex cover problem and the hitting set problem. The combination problem considered in this paper is to assign some of the jobs on m parallel machines such that the jobs correspond to a feasible solution of the covering problem and the objective is to minimize the makespan. We propose a unified approximation algorithm for the combination problem in which the parallel machines are uniformly related. An improved result is obtained when the parallel machines are identical. We apply our results to some specific combination problems to demonstrate our approach.