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SUBDIVIDING THE POLAR OF A FACE
, 708
"... ABSTRACT. Let S be a convex polytope. The set of all valid inequalities carries the structure of a convex polytope S △ , called the polar (polytope) of S. The facial structure of the polar provides information for each of its points: two points a and b are in the same face of S △ if and only if the ..."
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ABSTRACT. Let S be a convex polytope. The set of all valid inequalities carries the structure of a convex polytope S △ , called the polar (polytope) of S. The facial structure of the polar provides information for each of its points: two points a and b are in the same face of S △ if and only if the faces of S obtained by intersecting it with the hyperplanes given respectively by a and b coincide. Suppose now that S is a face of another polyhedron P. Then the points of S △ carry some additional information: the set of faces of P which one can obtain by “rotating” the hyperplane given by a point. This additional information is captured by the structure of a polyhedral complex subdividing S △. In this paper, we study this subdivision for the following examples: The Birkhoff polytope as a face of the matching polytope; the permutahedron as a facet of another permutahedron; the Symmetric Traveling Salesman Polytope, also known as Hamiltonian Cycle polytope, as a face of the connected Eulerian multigraph polyhedron, also known as Graphical Traveling Salesman Polyhedron. 1.
On the parsimonious property of relaxations of the symmetric traveling salesman polytope. arXiv/math.CO:blah, submitted
"... ABSTRACT. We relate the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope to a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron. This relationship is quite surprising. The proof is elegant and geometric: it makes use of recent re ..."
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ABSTRACT. We relate the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope to a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron. This relationship is quite surprising. The proof is elegant and geometric: it makes use of recent results on “flattening ” parts of the boundary complex of the polar of the Graphical Traveling Salesman Polyhedron. The Symmetric Traveling Salesman Polytope STSP(n) is the convex hull of all cycles (connected 2regular graphs) on a fixed vertex set V of cardinality n. The Graphical Traveling Salesman Polyhedron GTSP(n) is the convex hull of all connected Eulerian multigraphs a fixed vertex set on V. It contains STSP(n) as a face, defined by a certain system of linear equations. A relaxation is a system of linear inequalities which are facetdefining for STSP(n) and GTSP(n). It has the parsimonious property if, for a certain set of linear objective functions, the following holds: the minimum of this function over the relaxation does not increase when the above mentioned equations are added to the relaxation. 1.
ON THE FACIAL STRUCTURE OF SYMMETRIC AND GRAPHICAL TRAVELING SALESMAN POLYHEDRA
, 2009
"... The Symmetric Traveling Salesman Polytope Sn for a fixed number n of cities is a face of the corresponding Graphical Traveling Salesman Polyhedron Pn. This has been used to study facets of Sn using Pn as a tool. In this paper, we study the operation of “rotating” (or “lifting”) valid inequalities fo ..."
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The Symmetric Traveling Salesman Polytope Sn for a fixed number n of cities is a face of the corresponding Graphical Traveling Salesman Polyhedron Pn. This has been used to study facets of Sn using Pn as a tool. In this paper, we study the operation of “rotating” (or “lifting”) valid inequalities for Sn to obtain a valid inequalities for Pn. As an application, we describe a surprising relationship between (a) the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope and (b) a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron.