Approximating Interval Scheduling Problems with Bounded Profits

Abstract

We consider the Generalized Scheduling Within Intervals(GSWI) problem: given a set J of jobs and a set $\mathcal{I}$ of intervals, where each job j ∈ J has in interval I $\in \mathcal{I}$ 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 {0,1}-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 $I=I_{j} \in \mathcal{I}$ 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.