## Help me recognize classical problem in task

There is a task. For given lattice (isotropic discrete abelian group ) in R^3 for every vertice y let M(y) - set of neighbour vertices (not necessarily closest one's, for other vertice z: M(z)=M(y)+(z-y)={a+z=y| a \in M(y)}).
Find a mininmal with respect to cardinal subset B(y) of M(y) such that for every x \in M(y) :
B(y) \intersection B(x) isn't empty. (B(x)=B(y)+(x-y) - again isotropical translation).

It seems to be NP problem but in which cases it could be solved
or at least equivalent tasks couldn't be obtained.