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 J of intervals, where each job j ∈ J has in interval I ∈ I length (processing time) ℓ_j,I and profit P_{j, 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 [4] for the {0, l}-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 = I_j ∈ I with P_j,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.