On Some Polytopes Contained in the 0,1 Hypercube that Have a Small Chvátal Rank

Abstract

In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of infeasible 0,1 vectors that guarantee that P has a small Chvátal rank. Our conditions are in terms of the subgraph induced by these infeasible 0,1 vertices in the skeleton graph of the unit hypercube. In particular, we show that when this subgraph contains no 4-cycle, the Chvátal rank is at most 3; and when it has tree width 2, the Chvátal rank is at most 4. We also give polyhedral decomposition theorems when this graph has a vertex cutset of size one or two.