Efficient edge-skeleton computation for polytopes defined by oracles

Abstract: In general dimension, there is no known total polynomial algorithm for either convex hull or vertex enumeration, i.e. an algorithm whose complexity depends polynomially on the input and output sizes. It is thus important to identify problems (and polytope representations) for which total polynomial-time algorithms can be obtained. We offer the first total polynomial-time algorithm for computing the edge-skeleton (including vertex enumeration) of a polytope given by an optimization or separation oracle, where we are also given a superset of its edge directions. We also offer a space-efficient variant of our algorithm by employing reverse search. All complexity bounds refer to the (oracle) Turing machine model. There is a number of polytope classes naturally defined by oracles; for some of them neither vertex nor facet representation is obvious. We consider two main applications, where we obtain (weakly) total polynomial-time algorithms: Signed Minkowski sums of convex polytopes, where polytopes can be subtracted provided the signed sum is a convex polytope, and computation of secondary, resultant, and discriminant polytopes. Further applications include convex combinatorial optimization and convex integer programming, where we offer a new approach, thus removing the complexity's exponential dependence in the dimension.