Chvátal–Gomory cuts for the Steiner tree problem

The Steiner tree problem asks for a minimum weight subtree of an undirected graph spanning distinct terminal nodes. As one of the most fundamental network design problems, it has received much attention from both the mathematical programming and the algorithmic community and indeed, many integer linear programming (ILP) formulations of varying strength are known. The practical usefulness of the formulations can be enhanced with additional valid inequalities that reduce the gap between the efficiently computable continuous relaxation and the computationally difficult integer linear program. We aim to better understand the structure of these inequalities. In particular, we want to know how the inequalities can be obtained through Chvátal–Gomory cuts; a general procedure that is implemented in state-of-the-art ILP solver software. We start with the most basic ILP formulation for the Steiner tree problem and analyze how two known classes of strengthening inequalities – partition inequalities and odd-hole inequalities – can be derived systematically as Chvátal–Gomory cuts and how these inequalities themselves can be used to derive Chvátal–Gomory cuts. Along the way, we prove that the Chvátal rank of k-partition and k-odd-hole inequalities is at most k−2 and k−1, respectively. In particular, this proves that k iterations of the Chvátal–Gomory cut procedure suffice to obtain a given k+2-partition or a given k+1-odd-hole inequality.