On the geometric separability of Boolean functions Original Research Article

We investigate the complexity of the MEMBERSHIP problem for some geometrically defined classes of Boolean functions, i.e., the complexity of deciding whether a Boolean function given in DNF belongs to the class. We give a general argument implying that this problem is co-NP-hard for any class having some rather benign closure properties. Applying this result we show that the MEMBERSHIP problem is co-NP-complete for the class of linearly separable functions, threshold functions of order k (for any fixed k ⩾ 0), and some binary-parameter analogues of these classes. Finally, we obtain that the considered problem for unions of k ⩾ 3 halfspaces is NP-hard, co-NP-hard and belongs to Σ2p, and that the optimal threshold decomposition of a Boolean function as a union of halfspaces cannot even be efficiently approximated in a very strong sense unless P = NP. In some cases we improve previous hardness results on the considered problems.