Resource Scheduling with Supply Constraint and Linear Cost

Parallel resource and machine scheduling received a lot of attentions in the literature. In most of parallel scheduling models, the numbers of resources or machines that jobs must be assigned at each time unit are the inputs of the problems, i.e., jobs cannot be assigned to arbitrary numbers of resources or machines at the same time unit. In this paper, we consider the following scheduling problem. There are m resources available at each time unit, and n jobs, each requiring a number of resources. A resource can only be assigned to one job at each time unit while jobs can get arbitrary numbers of resources at the same time unit. The objective is to find a schedule that minimizes the total weighted completion time. We show this problem is NP-hard when the number of resource available at each time unit is the input of the problem. We then give a simple greedy algorithm to achieve 2-approximation. Finally, we present a dynamic programming algorithm to solve this problem when the number of different resources requirements is a constant.