Fukuda, KomeiLiebling, Thomas M.Margot, François2006-02-132006-02-132006-02-13199710.1016/0925-7721(95)00049-6https://infoscience.epfl.ch/handle/20.500.14299/222827In this paper, we investigate the applicability of backtrack technique to solve the vertex enumeration problem and the face enumeration problem for a convex polyhedron given by a system of linear inequalities. We show that there is a linear-time backtrack algorithm for the face enumeration problem whose space complexity is polynomial in the input size, but the vertex enumeration problem requires a backtrack algorithm to solve a decision problem, called the restricted vertex problem, for each output, which is shown to be NP-complete. Some related NP-complete problems associated with a system of linear inequalities are also discussed, including the optimal vertex problems for polyhedra and arrangements of hyperplanes.text::journal::journal article::research article