摘要：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 either the largest job size p_max, or the number of jobs n are polynomially bounded in the instance size I , there are algorithms with complexity I ^poly(k). Our main result is that this is unlikely to be improved, because Q C_max is W[1]-hard parameterized by k already when n, p_max, and the numbers describing the speeds are polynomial in I ; the same holds for R HM C_max (without speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives ð"â,,-norm minimization of the load vector and, partially, sum of weighted completion times â^' 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.
