The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines ( \(Q|HM|C_{\max }\) ) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model ( \(R|HM|C_{\max }\) ), we show that if the largest job size \(p_{\max }\) or the number of jobs n is polynomially bounded in the instance size |I|, there are algorithms with complexity \(|I|^{{{\,\mathrm{\textrm{poly}}\,}}(k)}\) . Our main result is that this is unlikely to be improved because \(Q||C_{\max }\) is \(\mathsf {W[1]}\) -hard parameterized by k already when n, \(p_{\max }\) , and the numbers describing the machine speeds are polynomial in |I|; the same holds for \(R||C_{\max }\) (without machine speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives \(\ell _2\) -norm minimization of the load vector and, partially, sum of weighted completion times \(\sum w_j C_j\) . Along the way, we answer affirmatively the question whether makespan minimization on identical machines ( \(P||C_{\max }\) ) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for \(Q||C_{\max }\) , this implies that the complexity of \(P|HM|C_{\max }\) is the only remaining open case.
