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].
Similar content being viewed by others
References
A. I. Barvinok: Combinatorial complexity of orbits in representations of the symmetric group,Advances in Soviet Mathematics, vol. 9, 1992, pp 161–182.
L. J. Billera, andA. Sarangarajan: The combinatorics of permutation polytopes, in L. Billera, C. Greene, R. Simion and R. Stanley, eds.,Formal Power Series and Algebraic Combinatorics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol 24, American Mathematical Society, Providence, RI, 1996.
R. Brualdi, andP. Gibson: Convex polyhedra of doubly stochastic matrices, I, II, III,Journal of Combin. Theory, A22 (1977), 192–230; B22 (1977), 175–198; A22 (1977), 338–351.
V. A. Emelichev, M. M. Kovalev, andM. K. Kravtsov:Polytopes, Graphs and Optimization, Cambridge University Press, New York, 1984, 211–227.
E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, andD. B. Shmoys, eds.:The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, New York, 1985.
L. L. Lovász, andM. D. Plummer:Matching Theory, Elsevier Science Pub. Co., New York, 1986.
M. R. Rao: Adjacency of the traveling salesman tours and 0–1 vertices,SIAM J. Appl. Math.,30, (1976), 191–198.
G. Ziegler:Lectures on Polytopes, Graduate Text in Mathematics 152, Springer-Verlag, New York, 1995. (Updates and corrections available athttp://winnie.math.tu-berlin.de/∼ziegler/.)
Author information
Authors and Affiliations
Additional information
Partially supported by NSF grant DMS-9207700.
Rights and permissions
About this article
Cite this article
Billera, L.J., Sarangarajan, A. All 0–1 polytopes are traveling salesman polytopes. Combinatorica 16, 175–188 (1996). https://doi.org/10.1007/BF01844844
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01844844