  We study the shared processor scheduling problem with a single shared
processor where a unit time saving (weight) obtained by processing a job on the
shared processor depends on the job. A polynomial-time optimization algorithm
has been given for the problem with equal weights in the literature. This paper
extends that result by showing an $O(n \log n)$ optimization algorithm for a
class of instances in which non-decreasing order of jobs with respect to
processing times provides a non-increasing order with respect to weights ---
this instance generalizes the unweighted case of the problem. This algorithm
also leads to a $\frac{1}{2}$-approximation algorithm for the general weighted
problem. The complexity of the weighted problem remains open.
