Comparative computational results for some vertex and facet enumeration codes

We report some computational results comparing parallel and sequential codes for vertex/facet enumeration problems for convex polyhedra. The problems chosen span the range from simple to highly degenerate polytopes. We tested one code (lrs) based on pivoting and four codes (cddr+, ppl , normaliz , porta) based on the double description method. normaliz employs parallelization as do the codes plrs and mplrs which are based on lrs . We tested these codes using various hardware configurations with up to 1200 cores. Major speedups were obtained by parallelization, particularly by the code mplrs which uses MPI and can operate on clusters of machines. 1 Background and polytopes tested A convex polyhedron P can be represented by either a list of vertices and extreme rays, called a Vrepresentation, or a list of its facet defining inequalities, called an H-representation. The vertex enumeration problem is to convert an H-representation to a V-representation. The computationally equivalent facet enumeration problem performs the reverse transformation. For further background see G. Ziegler [12]. In this note we consider only polytopes (bounded polyhedra) so extreme rays will not be required. Furthermore, for technical simplicity in this description, we assume that all polytopes are full dimensional. Neither condition is required for the algorithms tested and in fact some of our test problems are not full dimensional. The input for either problem is represented by an m by n matrix. For the vertex enumeration problem this is a list of m inequalities in n − 1 variables whose intersection define P . For a facet enumeration problem it is a list of the vertices of P each beginning with a 1 in column one. So in either case, under our assumption, the dimension of P is n− 1. One of the features of this type of enumeration problem is that the output size varies widely for given input parameters m and n. This is shown explicitly by McMullen’s Upper Bound Theorem (see, e.g., [12]) which is tight. It states that for a vertex enumeration problem with parameters m,n we have: