We study a resource allocation problem where jobs have the following characteristics: Each job consumes some quantity of a bounded resource during a certain time interval and induces a given profit. We aim to select a subset of jobs with maximal total profit such that the total resource consumed at any point in time remains bounded by the given resource capacity. While this case is trivially $\mathcal{NP}$-hard in general and polynomially solvable for uniform resource consumptions, our main result shows the $\mathcal{NP}$-hardness for the case of general resource consumptions but uniform profit values, i.e.\ for the case of maximizing the number of performed jobs. This result applies even for proper time intervals. We also give a deterministic (1/2-\eps)$-- approximation algorithm for the general problem on proper intervals improving upon the currently known $1/3$ ratio for general intervals. Finally, we study the worst-case performance ratio of a simple greedy algorithm.
Citation
The final version of this article appeared in Theoretical Computer Science: http://www.sciencedirect.com/science/article/pii/S0304397510004639 DOI: 10.1016/j.tcs.2010.08.028