| Abstract: |
Given a set of jobs, each consisting of a number of weighted intervals on the real line, and a number m of machines, we study the problem of selecting a maximum weight subset of the
intervals such that at most one interval is selected from each job and, for any point p on the real line, at most m intervals containing p are selected. This problem has
applications in molecular biology, caching, PCB assembly, and scheduling. We give a parameterized algorithm GREEDY(alpha) and show that there are values of the parameter alpha so that
GREEDY(alpha) produces a 1/2-approximation in the case of unit weights, a 1/8-approximation in the case of arbitrary weights, and a (3-2*sqrt(2))-approximation in the case where the weights of
all intervals corresponding to the same job are equal. Algorithm GREEDY(alpha) belongs to the class of "myopic" algorithms, which are algorithms that process the given intervals in order of
non-decreasing right endpoints and can either reject or select each interval (rejections are irrevocable). We use competitive analysis to show that GREEDY(alpha) is an optimal myopic algorithm in
the case of unit weights and in the case of equal weights per job, and is close to optimal in the case of arbitrary weights. |
