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ACImplementation of the Reverse Search Vertex Enumeration Algorithm.pdf

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ACImplementation of the Reverse Search Vertex Enumeration Algorithm

A C Implementation of the Reverse Search Vertex Enumeration Algorithm David Avis School of Computer Science McGill University 3480 University, Montreal, Quebec H3A 2A7 Revised June 1993 ABSTRACT This report documents a C implementation of the reverse search vertex enu- meration algorithm for convex polyhedra of Avis and Fukuda. The implementa- tion uses multiple precision rational arithmetic and contains a few improvements over the original algorithm, resulting in a small reduction in the complexity in most cases. For a polyhedron with n 0 inequalities in d non-negative variables and non-degenerate origin, it finds all bases in time O(n 0 d 2 ) per basis. It is also shown how the implementation can be improved to run in time O(n 0 d min(n 0 , d)) time per basis for any polyhedron. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex). Computational experience is included in the report. 1. Background A new enumeration method called Reverse Search was recently introduced by Avis and Fukuda[1] (hereafter referred to as AF) and applied to the problem of finding all vertices of a convex polyhedron. This report documents a C implementation of the algorithm. The implemen- tation uses multiple precision rational arithmetic and contains a few improvements over the origi- nal algorithm. It can handle problems of quite large size, especially for simple polyhedra. Compu- tational experience is included in the report. This implementation in C was motivated by an earlier implementation by K. Fukuda and I. Mizukoshi in Mathematica[5]. Their implementation demonstrates the simplicity and effec- tiveness of the method, and provides impressive graphical output (see, eg. Figure 3.5). However, due to the inherent limitations of Mathematica, it is only computationally feasible for small prob- lems. The C implementation described here runs several hundred times faster for the medium sized problems described