On switching paths polyhedra


A class of integer polyhedra with totally dual integral (tdi) systems is proposed, which generalizes and unifies the “Switching Paths Polyhedra” of Hoffman (introduced in his generalization of Max Flow-Min Cut) and such polyhedra as the convex hull of (the incidence vectors of) all “path-closed sets” of an acyclic digraph, or the convex hull of all sets partitionable intok path-closed sets. As an application, new min-max theorems concerning the mentioned sets are given. A general lemma on when a tdi system of inequalities is box tdi is also given and used.

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  1. [1]

    K. B. Cameron, Polyhedral and algorithmic ramifications of antichainsPh. D. Thesis (1982) Univ. of Waterloo, Waterloo, Ontario, Canada.

    Google Scholar 

  2. [2]

    R. P. Dilworth, A decomposition theorem for partially ordered sets,Ann. of Math. 51 (1950), 161–166.

    Article  MathSciNet  Google Scholar 

  3. [3]

    J. Edmonds andR. Giles, A min-max relation for submodular functions on graphs, (in: Studies in Integer Programming),Ann. of Discrete Math. 1 (1977), 185–204.

    MathSciNet  Article  Google Scholar 

  4. [4]

    T. Gallai andA. N. Milgram, Verallgemeinerung eines graphentheoretischen Satzes von Rédei,Acta Sc. Math. 21 (1960), 181–186.

    MATH  MathSciNet  Google Scholar 

  5. [5]

    C. Greene andD. Kleitman, The structure of Spernerk-families,J. Combinatorial Theory Ser. A 20 (1976), 80–88.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    H. Gröflin, Path-closed sets,Combinatorica 4 (1984), 281–290.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    H. Gröflin andA. J. Hoffman, On matroid intersection,Combinatorica 1 (1981), 43–47.

    MATH  MathSciNet  Google Scholar 

  8. [8]

    H. Gröflin andA. J. Hoffman, Lattice polyhedra II: generalization, constructions and examples,Ann. of Discrete Math. 15 (1982), 189–203.

    MATH  Google Scholar 

  9. [9]

    A. J. Hoffman, A generalization of max flow-min cut,Mathematical Programming 6 (1974), 352–359.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    A. J. Hoffman andD. E. Schwartz, On partitions of a partially ordered set,J. Combinatorial Theory Ser. B 23 (1977), 3–13.

    MATH  Article  MathSciNet  Google Scholar 

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Gröflin, H. On switching paths polyhedra. Combinatorica 7, 193–204 (1987). https://doi.org/10.1007/BF02579449

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  • 90 C 05
  • 90 B 10
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