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Abstract The Holt-Klee Condition states that there exist at least d vertex-disjoint strictly monotone paths from the source to the sink of a polytopal digraph consisting of the set of vertices and arcs of a polytope P directed by a linear objective function in general position. Studying these paths has the potential to provide new insight into a lo...
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Context 1
... our example, we show in Figure 5 a reverse search tree for f , as a subtree of the full reverse search tree for P , rooted at f . The dotted edges show the degenerate pivots performed by Algorithm 2. Note the tree is rooted at the lex-max cobasis for f , {6, 7, 8}, for which the corresponding basis is lex-min. ...
Context 2
... oracle examines the next edge, the pivot on x 7 and x 5 , to vertex c which is also inadmissible as (d, c) is reversed. The oracle's last option, pivot on x 9 and x 8 , is a nondegenerate pivotThe nondegenerate pivot x 6 for x 9 , to vertex d is inadmissible as d ∈ S. As no more nondegenerate pivots are possible from {6, 7, 8} the oracle initiates a reverse search on the lex-positive bases of f , setting the lex-max cobasis {6, 7, 8} as the root of a reverse search tree (refer to Figure 5). We construct an artificial objective function z r.s. ...
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Citations
... Reverse search tree of lex-positive bases rooted at(6,7,8) variable pivoted on in B. The adjacency oracle attempts to find the next admissible edge from B by incrementing i until either i > d, or the pivot on this column of the dictionary is nondegenerate. In the latter case, this nondegenerate pivot defines a vertex w such that vw is an edge of G(P ). ...
