Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \leq 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n-d \leq 6. We show this result by showing this bound for a more general structure -- so-called matroid polytopes -- by reduction to a small number of satisfiability problems.Comment: 9 pages; update shortcut constraint discussio

Recursive structure of S-matrices and an O(m2) algorithm for recognizing strong sign solvability

AbstractAn S-matrix is an m×(m+1) real matrix A such that for each matrix à of the same sign pattern as A, Ã's columns are the vertices of an m-simplex in Rm that has the origin in its interior. An S∗-matrix is one that can be made into an S-matrix by replacing some columns with their negatives. Such matrices are of interest because of their fundamental role in the study of sign solvability. New results on the recursive structure of these classes of matrices are presented here, and are used as the basis of algorithms of time complexity O(m2) for recognizing members of the classes and for testing the strong sign solvability of linear systems