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Computing monotone disjoint paths on polytopes

Author

Listed:
  • David Avis

    (McGill University)

  • Bohdan Kaluzny

    (McGill University)

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.

Suggested Citation

  • David Avis & Bohdan Kaluzny, 2008. "Computing monotone disjoint paths on polytopes," Journal of Combinatorial Optimization, Springer, vol. 16(4), pages 328-343, November.
    • Handle: RePEc:spr:jcomop:v:16:y:2008:i:4:d:10.1007_s10878-008-9151-3
    DOI: 10.1007/s10878-008-9151-3
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    References listed on IDEAS

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    1. Michael J. Todd, 1980. "The Monotonic Bounded Hirsch Conjecture is False for Dimension at Least 4," Mathematics of Operations Research, INFORMS, vol. 5(4), pages 599-601, November.
    2. Fukuda, K. & Terlaky, T., 1999. "On the existence of a short admissible pivot sequence for feasibility and linear optimization problems," Pure Mathematics and Applications, Department of Mathematics, Corvinus University of Budapest, vol. 10(4), pages 431-447.
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