On cutting-plane proofs in combinatorial optimization

Abstract Gomory's cutting-plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron. The number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron. We prove that polyhedra featured in popular formulations of the stable-set problem, the set-covering problem, the set-partitioning problem, the knapsack problem, the bipartite-subgraph problem, the maximum-cut problem, the acyclic-subdigraph problem, the asymmetric traveling-salesman problem, and the traveling-salesman problem have arbitrarily high rank. In particular, we prove conjectures of Barahona, Grotschel, and Mahjoub; Chvatal; Grotschel and Pulleyblank; and Junger.

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