Approximating Interval Scheduling Problems with Bounded Profits Israel Beniaminy * Zeev Nutov † Meir Ovadia ‡ Abstract 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) 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 [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 = 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. Key-words. Interval scheduling, Approximation algorithm. 1 Introduction We consider the following problem: Generalized Scheduling Within Intervals (GSWI): Instance: A set J of jobs and a set I of intervals, where each job j ∈ J has in interval I ∈I length ℓ j,I and profit p j,I , and each interval I ∈I is given by [r I ,d I ). Objective: Find a maximum profit feasible schedule. * ClickSoftware Technologies, israel@clicksoftware.com † The Open University of Israel, nutov@openu.ac.il ‡ The Open University of Israel, meiro@cadance.com 1