Windows scheduling of arbitrary-length jobs on multiple machines

The generalized windows scheduling problem for n jobs on multiple machines is defined as follows: Given is a sequence, I=〈(w1,ℓ1),(w2,ℓ2),…,(wn,ℓn)〉 of n pairs of positive integers that are associated with the jobs 1,2,…,n, respectively. The processing length of job i is ℓi slots where a slot is the processing time of one unit of length. The goal is to repeatedly and non-preemptively schedule all the jobs on the fewest possible machines such that the gap (window) between two consecutive beginnings of executions of job i is at most wi slots. This problem arises in push broadcast systems in which data are transmitted on multiple channels. The problem is NP-hard even for unit-length jobs and a (1+ε)-approximation algorithm is known for this case by approximating the natural lower bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W(I)=\sum_{i=1}^{n}(1/w_{i})$\end{document}. The techniques used for approximating unit-length jobs cannot be extended for arbitrary-length jobs mainly because the optimal number of machines might be arbitrarily larger than the generalized lower bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W(I)=\sum_{i=1}^{n}(\ell_{i}/w_{i})$\end{document}. The main result of this paper is an 8-approximation algorithm for the WS problem with arbitrary lengths using new methods, different from those used for the unit-length case. The paper also presents another algorithm that uses 2(1+ε)W(I)+logwmax machines and a greedy algorithm that is based on a new tree representation of schedules. The greedy algorithm is optimal for some special cases, and computational experiments show that it performs very well in practice.