Testing membership in the {0, 1/2}-closure is strongly NP-hard, even for polytopes contained in the n-dimensional 0/1-cube

Caprara and Fischetti introduced a class of cutting planes, called {0, 1/2}-cuts, which are valid for arbitrary integer linear programs. They also showed that the associated separation problem is strongly NPhard. We show that separation remains strongly NP-hard, even when all integer variables are binary, even when the integer linear program is a set packing problem, and even when the matrix of left-hand side coefficients is the clique matrix of a graph containing a small number of maximal cliques. In fact, we show these results for the membership problem, which is weaker than separation.