On transportation polytopes with the minimum number of k-faces

On transportation polytopes with the minimum number of k-faces -- Criteria of belonging of a non-degenerate transportation polytope with a given number of faces (of maximum dimension) to the dass of polytopes with the minimum number of fc-faces of all dimensions (beginning with zero)'are suggested. A formula for this number is obtained. 1. INTRODUCTION One of the basic problems of combinatorial theory of polyhedra (going back to Euler) deals with the description of the range of values of the vector function f(M) = (/o(M), /i(M),..., fd-i(M)) whose fcth component is equal to the number of Maces of a d-polyhedron M. Up to now this problem is solved only for the classes of d-polyhedra with the number of vertices not greater than d + 3, and also for polytopes of different combinatorial types: Simplexes, prisms, pyramids [1]. The problem of estimating the bounds for the variation of components of the vector f(M), provided that the other components are fixed, is also known (see [1, 2]). Most often the number of faces (of maximum dimension) is fixed, and one tries to obtain bounds for the other components. Such investigations are carried out both for abstract polyhedra and for the polyhedra of some combinatorial optimization problems. A criterion of belonging http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

On transportation polytopes with the minimum number of k-faces

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.2.115
Publisher site
See Article on Publisher Site

Abstract

-- Criteria of belonging of a non-degenerate transportation polytope with a given number of faces (of maximum dimension) to the dass of polytopes with the minimum number of fc-faces of all dimensions (beginning with zero)'are suggested. A formula for this number is obtained. 1. INTRODUCTION One of the basic problems of combinatorial theory of polyhedra (going back to Euler) deals with the description of the range of values of the vector function f(M) = (/o(M), /i(M),..., fd-i(M)) whose fcth component is equal to the number of Maces of a d-polyhedron M. Up to now this problem is solved only for the classes of d-polyhedra with the number of vertices not greater than d + 3, and also for polytopes of different combinatorial types: Simplexes, prisms, pyramids [1]. The problem of estimating the bounds for the variation of components of the vector f(M), provided that the other components are fixed, is also known (see [1, 2]). Most often the number of faces (of maximum dimension) is fixed, and one tries to obtain bounds for the other components. Such investigations are carried out both for abstract polyhedra and for the polyhedra of some combinatorial optimization problems. A criterion of belonging

Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

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