PTAS for a variant of multiple knapsack?
If I'm not mistaken, in the introduction of Bansal and Sviridenko, they mention that your special case - where children value items equally aka jobs take the same time on all machines - was the topic of Woeginger (1994) and shown to have a PTAS. Woeginger also points out that the problem is strongly NP-hard and hence does not permit an FPTAS unless P=NP.

Note however that if the number of bins $m$ is fixed rather than being part of the input, then the usual discretization method used for Knapsack and others, where items are rounded to powers of $(1+\epsilon/n)$, seems to yield an FPTAS.