Interval selection: Applications, algorithms, and lower bounds

Given a set of jobs, each consisting of a number of weighted intervals on the real line, and a positive integer m, 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. We give a parameterized algorithm GREEDYα that belongs to the class of "myopic" algorithms, which are deterministic algorithms that process the given intervals in order of non-decreasing right endpoint and can either reject or select each interval (rejections are irrevocable). We show that there are values of the parameter α so that GREEDYα produces a 2-approximation in the case of unit weights, an 8-approximation in the case of arbitrary weights, and a (3 + 2√2)- approximation in the case where the weights of all intervals corresponding to the same job are equal. We also show that no deterministic myopic algorithm can achieve ratio better than 2 in the case of unit weights, better than ≈ 7.103 in the case of arbitrary weights, and better than 3 + 2√2 in the case where the weights of all intervals corresponding to the same job are equal. Furthermore, we give additional results for the case where all intervals have the same length as well as a lower bound of e/e-1≈ 1.582 on the approximation ratio of randomized myopic algorithms in the case of unit weights.