Approximations for Throughput Maximization
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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. For m=1, Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation with ratio 1-1/e-ε (for any ε>0). Recently Im-Li-Moseley [IPCO2017] presented an algorithm with ratio 1-1/e-ε_0 for some absolute constant ε_0>0 for any fixed m. They also presented an algorithm with ratio 1-O(√(log m/m))-ε for general m which approaches 1 as m grows. The approximability of the problem for m=O(1) remains a major open question. In this paper we 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 (1-ε)-approximation that runs in time O(n^mc^7ε^-6log T), where T is the largest deadline. For constant m and constant c this yields a PTAS. As a corollary, we obtain a QPTAS for when p_j's come from a set of size Poly(n) and m= Poly(log n) with (1+ε)-speed up of machines. More specifically, if each value p_j is bounded by Poly(n) (but no bound on the number of p_j's) for any ε>0, there is an algorithm that finds a (1-ε)-approximate solution on (1+ε)-speed machines; the algorithm runs in time O(n^ε^-13mlog^7 n).
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