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. The study of paths on polytopal digraphs stems from a long standing problem, that of designing a polynomial-time pivot method, or proving none exists. To study disjoint paths it would be useful to have a tool to compute them. Without explicitly computing the digraph we develop an algorithm to compute a maximum cardinality set of source to sink paths in a polytope, even in the presence of degeneracy. The algorithm uses a combination of networks flows, the simplex method, and reverse search. An implementation is available.
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Avis, D., Kaluzny, B. Computing monotone disjoint paths on polytopes. J Comb Optim 16, 328–343 (2008). https://doi.org/10.1007/s10878-008-9151-3
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DOI: https://doi.org/10.1007/s10878-008-9151-3