In this paper we consider the multidimensional binary vector assignment problem. An input of this problem is defined by m disjoint multisets V , V , . . . , V , each composed of n binary vectors of size p. An output is a set of n disjoint m-tuples of vectors, where each m-tuple is obtained by picking one vector from each multiset V . To each m-tuple we associate a p dimensional vector by applying the bit-wise AND operation on the m vectors of the tuple. The objective is to minimize the total number of zeros in these n vectors. We denote this problem by min ∑ 0, and the restriction of this problem where every vector has at most c zeros by (min ∑ 0)#0≤c. (min ∑ 0)#0≤2 was only known to beAPX-complete, even for m = 3 . We show that, assuming the unique games conjecture, it is NP-hard to (n− ε)-approximate (min ∑ 0)#0≤1 for any fixed n and ε. This result is tight as any solution is a n-approximation. We also prove without assuming UGC that (min ∑ 0)#0≤1 is APX-complete even for n = 2, and we provide an example of n − f(n,m)-approximation algorithm for min ∑ 0. Finally, we show that (min ∑ 0)#0≤1 is polynomialtime solvable for fixed m (which cannot be extended to (min ∑ 0)#0≤2 according to ).