  In this paper, we consider the problem of choosing a minimum cost set of
resources for executing a specified set of jobs. Each input job is an interval,
determined by its start-time and end-time. Each resource is also an interval
determined by its start-time and end-time; moreover, every resource has a
capacity and a cost associated with it. We consider two versions of this
problem. In the partial covering version, we are also given as input a number
k, specifying the number of jobs that must be performed. The goal is to choose
k jobs and find a minimum cost set of resources to perform the chosen k jobs
(at any point of time the capacity of the chosen set of resources should be
sufficient to execute the jobs active at that time). We present an O(log
n)-factor approximation algorithm for this problem.
  We also consider the prize collecting version, wherein every job also has a
penalty associated with it. The feasible solution consists of a subset of the
jobs, and a set of resources, to perform the chosen subset of jobs. The goal is
to find a feasible solution that minimizes the sum of the costs of the selected
resources and the penalties of the jobs that are not selected. We present a
constant factor approximation algorithm for this problem
