On optimality of exact and approximation algorithms for scheduling problems

We consider the classical scheduling problem on parallel identical machines to minimize the makespan. Under the exponential time hypothesis (ETH), lower bounds on the running times of exact and approximation algorithms are characterized. We achieve the following results: (1) For scheduling on a constant number m of identical machines, denoted by Pm||Cmax, a fully polynomial time approximation scheme (FPTAS) of running time (1/ )O(m )|I|O(1) for any constant δ > 0 implies that ETH fails (where |I| is the length of the input). It follows that the best-known FPTAS of running time O(n) + (m/ )O(m) for the more general problem with a constant number m of unrelated machines Rm||Cmax is essentially the best possible. (2) For scheduling on an arbitrary number of identical machines, denoted by P ||Cmax, a polynomial time approximation scheme (PTAS) of running time 2O((1/ ) )|I|O(1) for any δ > 0 also implies that ETH fails. Thus the bestknown PTAS of running time 2O(1/ 2 log(1/ )) + O(n log n) is almost best possible in terms of running time. (3) For P ||Cmax, 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 2O(n 1−δ) for any δ > 0 implies that ETH fails. Thus the traditional dynamic programming algorithm of running time 2O(n) is essentially the best possible.