On the Rational Polytopes with Chvátal Rank 1

In this paper, we study the following problem: given a polytope P with Chvátal rank 1, does P contain an integer point? Boyd and Pulleyblank observed that this problem is in the complexity class NP ∩ co-NP, an indication that it is probably not NP-complete. We solve this problem in polynomial time for polytopes arising from the satisfiability problem of a given formula with at least three literals per clause, for simplices whose integer hull can be obtained by adding at most a constant number of Chvátal inequalities, and for rounded polytopes. We prove that any closed convex set whose Chvátal closure is empty has an integer width of at most n, and we give an example showing that this bound is tight within an additive constant of 1. The promise that a polytope has Chvátal rank 1 seems hard to verify though. We prove that deciding emptiness of the Chvátal closure of a given rational polytope P is NP-complete, even when P is contained in the unit hypercube or is a rational simplex, and even when P does not contain any integer point. This has two implications: (i) It is NP-hard to decide whether a given rational polytope P has Chvátal rank 1, even when P is contained in the unit cube or is a rational simplex; (ii) The optimization and separation problems over the Chvátal closure of a given rational polytope contained in the unit hypercube or of a given rational simplex are NP-hard. These results improve earlier complexity results of Cornuéjols and Li and Eisenbrand. Finally, we prove that, for any positive integer k, it is NP-hard to decide whether adding at most k Chvátal inequalities is sufficient to describe the integer hull of a given rational polytope.