Approximating Interval Scheduling Problems with Bounded Profits

We consider the Generalized Scheduling Within Intervals (GSWI) problem: given a set J of jobs and a set I of intervals, where each job j ∈ J has in interval I ∈ I length (processing time) ℓj,I and profit pj,I, find the highest-profit feasible schedule. The best approximation ratio known for GSWI is (1/2 − ε). We give a (1 − 1/e − ε)approximation scheme for GSWI with bounded profits, based on the work by Chuzhoy, Rabani, and Ostrovsky [5], for the {0, 1}-profit case. We also consider the Scheduling Within Intervals (SWI) problem, which is a particular case of GSWI where for every j ∈ J there is a unique interval I = Ij ∈ I with pj,I> 0. We prove that SWI is (weakly) NP-hard even if the stretch factor (the maximum ratio of job’s interval size to its processing time) is arbitrarily small, and give a polynomial-time algorithm for bounded profits and stretch factor < 2. Key-words. Interval scheduling, Approximation algorithm