Lexico-smallest Representations, Duality and Matching Polyhedra

Every convex polyhedron in Rd admits both Hand V-representations. In both types, a representation is canonical if it is minimal and unique up to some elementary operations. In this paper, we show the duality of canonical representations and that the canonical V-representations coincide with certain canonical H-representations. Also, we show how the lexico-smallest representation, a computationally convenient alternative to the usual orthogonal representation, can be computed efficiently. Finally, we illustrate our results by considering H-representations of the perfect matching polytope. In particular, we show that using the properties of the underlying graph results in sensible improvements in the running time of the computation of canonical representation.