On the optimality of approximation schemes for the classical scheduling problem

We consider the classical scheduling problem on parallel identical machines to minimize the makespan. There is a long history of studies on this problem, focusing on exact and approximation algorithms, and it is thus natural to consider whether these algorithms are optimal in terms of the running time. Under the exponential time hypothesis (ETH), we achieve the following results in this paper: • The scheduling problem on a constant numberm of identical machines, denoted by Pm||Cmax, is known to admit a fully polynomial time approximation scheme (FPTAS) of running time O(n) + (1/ ) (indeed, the algorithm works for an even more general problem where machines are unrelated). We prove this algorithm is essentially the best possible in the sense that a (1/ ) 1−δ) + n time PTAS implies that ETH fails. • The scheduling problem on an arbitrary number of identical machines, denoted by P ||Cmax, is known to admit a polynomial time approximation scheme (PTAS) of running time 2 2 log(1/ )) +O(n). We prove this algorithm is nearly optimal in the sense that a 2 ) 1−δ) +n time PTAS for any δ > 0 implies that ETH fails, leaving a small room for improvement. • The traditional dynamic programming algorithm for P ||Cmax is known to run in 2 time. We prove this is essentially the best possible in the sense that even if we restrict that there are n jobs and the processing time of each job is bounded by O(n), an exact algorithm of running time 2 1−δ) for any δ > 0 implies that ETH fails. To obtain our results we will provide two new reductions from 3SAT, one for Pm||Cmax and one for P ||Cmax. Indeed, the new reductions explore the structure of scheduling problems and can also lead to other interesting results. For example, the recent paper of Bhaskara et al. [2] consider the minimum makespan scheduling problem where the matrix of job processing times P = (pij)m×n is of a low rank. They prove that rank 4 scheduling is APX-hard while the rank 2 scheduling is not, leaving the classification of rank 3 scheduling as an open problem. Using the framework of our reduction for P ||Cmax, rank 3 scheduling is proved to be APX-hard [5].