Peter M. Gibson

TL;DR: The permanent function is used to determine geometrical properties of the set Ω n of all n × n nonnegative doubly stochastic matrices to find k-dimensional faces with at least 2k−1 + 1 vertices.

TL;DR: In this paper, a connection between bounded faces of doubly stochastic polyhedra and faces of transportation polytopes is established, and it is shown that there exists an absolute bound for the number of extreme points of d-dimensional bounded faces.

TL;DR: Properties of the graph G(Ω n ) of the polytope Ω n of all n × n nonnegative doubly stochastic matrices are studied and the number of prime graphs in any prime factor decomposition of G ( F ) equals theNumber of connected components of the neighborhood of any vertex of G( F ).

TL;DR: Affine and combinatorial properties of the polytope Ω n of all n × n nonnegative doubly stochastic matrices are investigated and it is found that if F is a face of Ωn of dimension d > 2, then F has at most 3( d −1) facets.

TL;DR: The a s s i g n m e n t p o l y t o p e g~, cons i s t s of all n x n n o n n e g a t i v e d o u b l y s tochas t i c ma t r i c e s , tha t is.