On the lattice programming gap of the group problems

Given a full-dimensional lattice Λ⊂ZkΛ⊂Zk and a cost vector l∈Q>0k, we are concerned with the family of the group problems equation(0.1)min{l⋅x:x≡r(modΛ),x≥0},r∈Zk. The lattice programming gap  gap(Λ,l) is the largest value of the minima in (0.1) as rr varies over ZkZk. We show that computing the lattice programming gap is NP-hard when kk is a part of input. We also obtain lower and upper bounds for gap(Λ,l) in terms of ll and the determinant of ΛΛ.