Partial monotonizations of Hamiltonian cycle polytopes: dimensions and diameters

Abstract

In this paper we study partial monotonizations and level polytopes of the Hamiltonian Cycle Polytope, also called the symmetric Traveling Salesman Polytope. The kth Level Polytope is the convex hull of the characteristic vectors corresponding to sets of k edges in K n that can be extended to Hamiltonian cycles ( n⩾3). For 0⩽ α⩽ k, the α-monotonization of the kth Level Polytope is the convex hull of the characteristic vectors corresponding to sets of at least α and at most k edges in K n that can be extended to Hamiltonian cycles ( n⩾3). It is shown that for α<k, α -monotonizations of level polytopes are full dimensional. We give upper and lower bounds for the diameters of the α-monotonizations and determine the number of 0-faces of the level polytopes and α-monotonizations. The main result of this paper is a proof that the diameter of the monotone Hamiltonian Cycle Polytope and the monotone Hamiltonian Path Polytope are each θ( log n) .