Capacitated Max-Batching with interval graph compatibilities

We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the weight of a clique is the maximum weight of any interval in the clique. The goal is then to find such a partition of minimum total weight. This graph problem can also be interpreted as a batch scheduling problem. Solving a long-standing open question, we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS for this case.