Stefano Picozzi
Convex polyhedra are important objects in various areas of mathematics and other disciplines. A fundamental result, known as Minkowski-Weyl theorem, states that every polyhedron admits two types of representations, either as the solution set to a finite system of linear inequalities or the Minkowski sum of a finite set of points and half-rays. These are usually referred to as its H-representations and its V-representations, respectively. Neither H-representations nor V-representations are unique. Hence, deciding whether two H-representations, or two V-representations describe the same polyhedron is a nontrivial problem. We identify particular representations which are easy to determine, are compact, and reflect the geometrical properties of the underlying polyhedron. Key ingredients to this discussion are affine transformations of polyhedra, duality and complexity of polyhedral decision problems. In this dissertation, we discuss in detail the problem of refining the definitions of Hand V- representations such as to guarantee a one to one correspondence between polyhedra and their representations. As a convenient formalism, we introduce the notion of the canonical representation rule, which is any function assigning to each polyhedron P a particular H-representation, and a particular V-representation. The orthogonal representation rule is a well-known example of such a function. We define the lexico-smallest representation rule, an improved canonical representation rule that produce representations which are nonredundant, make the underlying geometry transparent and, furthermore, are sparse. These rules also exhibit nice polarity properties that can be exploited for computation. The key computational tool is linear programing. Furthermore, we show that the lexico-smallest H-representation of a perfect bipartite matching polyhedron is easy to determine using the combinatorial properties of the underlying graph. Finally, we discuss the complexity of various polyhedral verification problems related to representation of convex polyhedra. The standard technique to identify the redundancies within an H- or a V-representation is to use linear programming. We discuss the complexity of the converse reduction. More precisely, we show that deciding the feasibility of a system of linear inequalities can be reduced to deciding redundancy in an H- or in a V- representation in linear time. Also, we will study the equivalence of these problems with those of identifying the implicit linearities within an H- or a V- representation, and the boundedness of a polyhedron.
EPFL2005
