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, and achieve the following results under the Exponential Time Hypothesis (ETH)

1. The scheduling problem on a constant number $m$ of identical machines, which is denoted as $Pm\mathrm{\mid }\mathrm{\mid }{C}_{max}$, is known to admit a fully polynomial time approximation scheme (FPTAS) of running time $O(n)+(1\mathrm{/}ϵ{)}^{O(m)}$ (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 $(1\mathrm{/}ϵ{)}^{O(m^{1-\delta })}+n^{O(1)}$ time FPTAS for any $\delta >0$ implies that ETH fails.
2. The scheduling problem on an arbitrary number of identical machines, which is denoted as $P\mathrm{\mid }\mathrm{\mid }{C}_{max}$, is known to admit a polynomial time approximation scheme (PTAS) of running time $2^{O(1\mathrm{/}{ϵ}^2{\log }^3(1\mathrm{/}ϵ))}+n^{O(1)}$. We prove this algorithm is nearly optimal in the sense that a $2^{O((1\mathrm{/}ϵ{)}^{1-\delta })}+n^{O(1)}$ time PTAS for any $\delta >0$ implies that ETH fails, leaving a small room for improvement. To obtain these results we will provide two new reductions from 3SAT, one for $Pm\mathrm{\mid }\mathrm{\mid }{C}_{max}$ and another for $P\mathrm{\mid }\mathrm{\mid }{C}_{max}$. Indeed, the new reductions explore the structure of scheduling problems and can also lead to other interesting results. For example, using the framework of our reduction for $P\mathrm{\mid }\mathrm{\mid }{C}_{max}$, Chen et al. (arXiv:1306.3727) is able to prove the APX-hardness of the scheduling problem in which the matrix of job processing times $P=(p_{ij}{)}_{m\times n}$ is of rank 3, solving the open problem mentioned by Bhaskara et al. (SODA 2013).