Approximating Interval Scheduling Problems with Bounded Profits

摘要

We consider the Generalized Scheduling Within Intervals (GSWI) problem: given a set J of jobs and a set T 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.