All 0–1 polytopes are traveling salesman polytopes

Abstract

We study the facial structure of two important permutation polytopes in\(\mathbb{R}^{n^2 } \), theBirkhoff orassignment polytopeB n , defined as the convex hull of alln×n permutation matrices, and theasymmetric traveling salesman polytopeT n , defined as the convex hull of thosen×n permutation matrices corresponding ton-cycles. Using an isomorphism between the face lattice ofB n and the lattice of elementary bipartite graphs, we show, for example, that every pair of vertices ofB n is contained in a cubical face, showing faces ofB n to be fairly special 0–1 polytopes. On the other hand, we show thatevery 0–1d-polytope is affinely equivalent to a face ofT n , 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].

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Partially supported by NSF grant DMS-9207700.

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Billera, L.J., Sarangarajan, A. All 0–1 polytopes are traveling salesman polytopes. Combinatorica 16, 175–188 (1996). https://doi.org/10.1007/BF01844844

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Mathematics Subject Classification (1991)

  • Primary: 52 B 05
  • 52 B 12
  • Secondary: 90 C 27