2-Lattice Polyhedra: Duality

This is the first in a series of papers that explores a class of polyhedra we call 2-lattice polyhedra. 2-Lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhderon is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice polyhedra, called matching 2-lattice polyhedra, that include all of the mentioned special cases except the intersection of two polymatroids, we characterize the largest member in the family of minimum covers in terms of the maximum “cardinality” vectors in the polyhedron. In fact, we show that this same characterization arises from considering only the extreme maximum cardinality vectors. This characterization is at the heart of our extreme point algorithm [3] for finding a maximum cardinality vector in a matching 2-lattice polyhedron.