Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper, we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (facet, respectively) enumeration problem is the facet (vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper, we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal--dual ...

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In this paper, we discuss the computational complexity of the following enumeration problem: Given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P . It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of P-enumerability introduced by Valiant [10]. The concept of strong P-enumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0/1-polytopes. Our implementation based on the commercial LP solver CPLEX 1 is superior to general vertex enumeration algorithms. We give an example how simplifications of...

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We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P = NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains

Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...

A polytope is the bounded intersection of a finite set of halfspaces of R d . Every polytope can also be represented as the convex hull conv V of its vertices (or extreme points) V . The convex hull problem is to convert from the vertex representation to the halfspace representation or (equivalently by geometric duality) vice-versa. Given an ordering v 1 : : : vn of the input vertices, after some initialization an incremental convex hull algorithm constructs halfspace descriptions Hn\Gammak : : : Hn where H i is the halfspace description of convf v 1 : : : v i g. Let m i denote jH i j, and let m denote mn . Let OE(d) denote d=d p d e \Gamma 1; in this paper we give families of polytopes for which mn\Gamma1 2 \Omega\Gamma m OE(d) ) for any ordering of the input. We also give a family of 0=1-polytopes with a similar blowup in intermediate size. Since mn\Gamma1 is not bounded by any polynomial in m, n, and d, incremental convex hull algorithms cannot in any reasonable sense be con...

We give an algorithm that constructs the Hasse diagram of the face lattice of a convex polytope P from its vertex-facet incidences in time O(minfn; mg '), where n is the number of vertices, m is the number of facets, is the number of vertex-facet incidences, and ' is the total number of faces of P . This improves results of Fukuda and Rosta [4], who described an algorithm for enumerating all faces of a d-polytope P in O minfn; mg d ' 2 steps. For simple or simplicial d-polytopes our algorithm can be specialized to run in time O(d '). Furthermore,

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