Combinatorica

, Volume 16, Issue 2, pp 175–188

All 0–1 polytopes are traveling salesman polytopes

  • Louis J. Billera
  • A. Sarangarajan
Article

DOI: 10.1007/BF01844844

Cite this article as:
Billera, L.J. & Sarangarajan, A. Combinatorica (1996) 16: 175. doi:10.1007/BF01844844

Abstract

We study the facial structure of two important permutation polytopes in\(\mathbb{R}^{n^2 } \), theBirkhoff orassignment polytopeBn, defined as the convex hull of alln×n permutation matrices, and theasymmetric traveling salesman polytopeTn, defined as the convex hull of thosen×n permutation matrices corresponding ton-cycles. Using an isomorphism between the face lattice ofBn and the lattice of elementary bipartite graphs, we show, for example, that every pair of vertices ofBn is contained in a cubical face, showing faces ofBn to be fairly special 0–1 polytopes. On the other hand, we show thatevery 0–1d-polytope is affinely equivalent to a face ofTn, ford∼logn, by showing that every 0–1d-polytope is affinely equivalent to the asymmetric traveling salesman polytope of some directed graph withn nodes. The latter class of polytopes is shown to have maximum diameter [n/3].

Mathematics Subject Classification (1991)

Primary: 52 B 0552 B 12Secondary: 90 C 27

Copyright information

© Akadémiai Kiadó 1996

Authors and Affiliations

  • Louis J. Billera
    • 1
  • A. Sarangarajan
    • 2
  1. 1.Department of Mathematics and School of Operations ResearchCornell UniversityIthacaUSA
  2. 2.Center for Applied MathematicsCornell UniversityIthacaUSA