Approximations for Throughput Maximization

In this paper we study the classical problem of throughput maximization. In this problem we have a collection J of n jobs, each having a release time \(r_j\), deadline \(d_j\), and processing time \(p_j\). They have to be scheduled non-preemptively on m identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their \([r_j,d_j]\) window. This problem has been studied extensively (even for the case of \(m=1\)). Several special cases of the problem remain open. Bar-Noy et al. (Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1–4, 1999, Atlanta, Georgia, USA, pp. 622–631. ACM, 1999, https://doi.org/10.1145/301250.301420) presented an algorithm with ratio \(1-1/(1+1/m)^m\) for m machines, which approaches \(1-1/e\) as m increases. For \(m=1\), Chuzhoy et al. (42nd Annual Symposium on Foundations of Computer Science (FOCS) 2001, 14–17 October 2001, Las Vegas, Nevada, USA, pp. 348–356. IEEE Computer Society, 2001) presented an algorithm with approximation with ratio \(1-\frac{1}{e}-\varepsilon \) (for any \(\varepsilon >0\)). Recently Im et al. (SIAM J Discrete Math 34(3):1649–1669, 2020) presented an algorithm with ratio \(1-1/e+\varepsilon _0\) for some absolute constant \(\varepsilon _0>0\) for any fixed m. They also presented an algorithm with ratio \(1-O(\sqrt{\log m/m})-\varepsilon \) for general m which approaches 1 as m grows. The approximability of the problem for \(m=O(1)\) remains a major open question. Even for the case of \(m=1\) and \(c=O(1)\) distinct processing times the problem is open (Sgall in: Algorithms - ESA 2012 - 20th Annual European Symposium, Ljubljana, Slovenia, September 10–12, 2012. Proceedings, pp 2–11, 2012). In this paper we study the case of \(m=O(1)\) and show that if there are c distinct processing times, i.e. \(p_j\)’s come from a set of size c, then there is a randomized \((1-{\varepsilon })\)-approximation that runs in time \(O(n^{mc^7{\varepsilon }^{-6}}\log T)\), where T is the largest deadline. Therefore, for constant m and constant c this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning.