Abstract
A class of polytopes is defined which includes the polytopes related to the assignment problem, the edge-matching problem on complete graphs, the multi-dimensional assignment problem, and many other set partitioning problems. Modifying some results due to Balas and Padberg, we give a constructive proof that the diameter of these polytopes is less than or equal to two. This result generalizes a result obtained by Balinski and Rusakoff in connection with the assignment problem.
Furthermore, it is shown that the polytope associated with the travelling salesman problem has a diameter less than or equal to two. A weaker form of the Hirsch conjecture is also shown to be true for this polytope.
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Padberg, M.W., Rao, M.R. The travelling salesman problem and a class of polyhedra of diameter two. Mathematical Programming 7, 32–45 (1974). https://doi.org/10.1007/BF01585502
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DOI: https://doi.org/10.1007/BF01585502