Is there a name for this generalized assignment problem variant with an additional indicator constraint

I have an optimization problem that's closely related to the generalized assignment problem with an additional constraint. Each knapsack $$i$$ has capacity $$b^i$$. There is a number of tasks $$j$$, each has a cost $$c_{i,j}$$ and a weight $$w_{i,j}$$ when assigned t knapsack $$i$$. The problem is to find an assignment $$A^*$$ such that the total cost is minimized while each knapsack's capacity is not exceeded by the items' weights. Given that the set of tasks are partitioned into a finite number of non overlapping $$l$$ groups (i.e., a group of tasks constitutes a job). The additional constraint is as follows: the assignment $$A^*$$ has a predefined integer $$\beta$$ of fully assigned groups of tasks: $$\sum_{\forall l }\sum_{\forall j} 1_{A*}(x_{i,j}) = \beta$$. I am well aware of the classic GAP and heuristics for it. Is there a name for this type of variant of GAP? What are some good resources to go about solving the problem with the additional indicator constraint?

P.S.: I am able to generate the exact solution, I am looking for good resources for approxaimations/heuristics