Deciding emptiness of the Gomory-Chvátal closure is NP-complete, even for a rational polyhedron containing no integer point

Abstract

Gomory-Chvátal cuts are prominent in integer programming. The Gomory-Chvátal closure of a polyhedron is the intersection of all half spaces defined by its Gomory-Chvátal cuts. In this paper, we show that it is NP-complete to decide whether the Gomory-Chvátal closure of a rational polyhedron is empty, even when this polyhedron contains no integer point. This implies that the problem of deciding whether the Gomory-Chvátal closure of a rational polyhedron P is identical to the integer hull of P is NP-hard. Similar results are also proved for the {−1, 0, 1}-cuts and {0, 1}-cuts, two special types of Gomory-Chvátal cuts with coefficients restricted in {−1, 0, 1} and {0, 1}, respectively.