Scheduling with non-renewable resources: minimizing the sum of completion times

We consider single-machine scheduling with a non-renewable resource. In this setting, we are given a set of jobs, each characterized by a processing time, a weight, and a resource requirement. At fixed points in time, certain amounts of the resource are made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that each job has enough resource available at the start of its processing. The objective that we consider is the minimization of the sum of weighted completion times. The main contribution of the paper is a PTAS for the case of 0 processing times (1|rm=1,pj=0|∑wjCj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1|rm=1,p_j=0|\sum w_jC_j$$\end{document}). In addition, we show strong NP-hardness of the case of unit resource requirements and weights (1|rm=1,aj=1|∑Cj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1|rm=1,a_j=1|\sum C_j$$\end{document}), thus answering an open question of Györgyi and Kis. We also prove that the schedule corresponding to the Shortest Processing Time First ordering provides a 3/2-approximation for the latter problem. Finally, we investigate a variant of the problem where processing times are 0 and the resource arrival times are unknown. We present a (4+ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(4+\epsilon )$$\end{document}-approximation algorithm, together with a (4-ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(4-\varepsilon )$$\end{document}-inapproximability result, for any ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}.