Journal ArticleDOI
Adjacency of the Traveling Salesman Tours and $0 - 1$ Vertices
TLDR
In this paper, a necessary and sufficient condition for adjacency on the convex hull of 0-1 feasible points was given for the set partitioning problem, and a strong bound was derived for the diameter of the polytope associated with the conveX hull of $0 -1$ feasible points.Abstract:
A necessary and sufficient condition is given for adjacency on the convex hull of 0–1 feasible points. The class of problems for which this condition is valid includes the set partitioning problem. A strong bound is derived for the diameter of the polytope associated with the convex hull of $0 - 1$ feasible points. A counterexample is given to a published necessary and sufficient condition for nonadjacency of two traveling salesman tours on their convex hull. A necessary condition is obtained for two tours to be nonadjacent on their convex hull. A sufficient condition for nonadjacency is also given. Examples are provided for the traveling salesman problem to show that neither the necessary condition is sufficient nor the sufficient condition is necessary. Finally, some adjacency properties are given for the traveling salesman tours on the assignment polytope.read more
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Journal ArticleDOI
The adjacency relation on the traveling salesman polytope is NP-Complete
TL;DR: This problem of determining whether two traveling salesman tours correspond to non-adjacent vertices of the convex polytope associated with the traveling salesman problem is shown to be NP-Complete for both the symmetric and nonsymmetric traveled salesman problem.
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All 0-1 Polytopes are Traveling Salesman Polytopes
TL;DR: It is shown thatevery 0–1d-polytope is affinely equivalent to a face of Tn, ford∼logn, by showing that every 0– 1d-Polytopes is affinably equivalent to the asymmetric traveling salesman polytope of some directed graph withn nodes.
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Adjacency on polymatroids
TL;DR: This paper characterizes adjacency for extreme points of a polymatroid by developing a polynomial algorithm that generates and lists all extreme points adjacent to a given extreme point of apolymatroid.
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The skeleton of the symmetric Traveling Salesman Polytope
TL;DR: It is shown that this skeleton contains a Hamiltonian tour such that the Hamiltonian cycles in K(n) corresponding to two successive vertices differ in a single interchange, i.e., the interchange graph corresponding to the TSP-polytope is Hamiltonian.
Journal ArticleDOI
On pedigree polytopes and Hamiltonian cycles
TL;DR: A polynomial time algorithm is given for nonadjacency testing in the pedigree polytope, whereas the corresponding problem is known to be NP-complete for Q"n.