Discrete & Computational Geometry

, Volume 20, Issue 3, pp 333–357 | Cite as

Primal—Dual Methods for Vertex and Facet Enumeration

  • D. Bremner
  • K. Fukuda
  • A. Marzetta

Abstract.

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 (resp. vertex) to the vertex (resp. halfspace) representation is called vertex enumeration (resp. 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 (resp. facet) enumeration problem is the facet (resp. 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 algorithms use a solution to the easy direction as an oracle to help solve the seemingly hard direction.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© 1998 Springer-Verlag New York Inc.

Authors and Affiliations

  • D. Bremner
    • 1
  • K. Fukuda
    • 2
  • A. Marzetta
    • 4
  1. 1.Department of Mathematics, University of Washington, Seattle, WA 98195, USA bremner@math.washington.eduUS
  2. 2.Department of Mathematics, Swiss Federal Institute of Technology, Lausanne, Switzerland CH
  3. 3.Institute for Operations Research, Swiss Federal Institute of Technology, Zurich, Switzerland fukuda@ifor.math.ethz.chCH
  4. 4.Institute for Theoretical Computer Science, Swiss Federal Institute of Technology, Zurich, Switzerland marzetta@inf.ethz.chCH

Personalised recommendations