An improved algorithm for parallel machine scheduling under additional resource constraints

We consider a classic machine scheduling problem under resource constraints. Given m parallel identical machines and a collection of additional but renewable resources, each task requires both a machine and one particular resource at any time of its processing. The goal is to allocate the machines and resources to the tasks so as to minimize the makespan, that is, the maximum completion time of all tasks. Because of NP-hardness of the problem, we design a three-stage combinatorial algorithm with performance ratio 2m+2m+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{2m+2}{m+3}$$\end{document} that improves the best prior ratio for m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 3$$\end{document} machines and ties with that for m=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=2$$\end{document} machines.