Path-closed sets


Given a digraphG = (V, E), call a node setTV path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR v, using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.

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  1. [1]

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

    Article  MathSciNet  Google Scholar 

  2. [2]

    J. Edmonds, Matroids and the greedy algorithm,Mathematical Programming 1 (1971), 127–136.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

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

    MathSciNet  Google Scholar 

  4. [4]

    D. R. Fulkerson, Blocking and Antiblocking Polyhedra,Mathematical Programming 1 (1971), 168–194.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    D. R. Fulkerson, Anti-blocking polyhedra,Journal of Combinatorial Theory 12 (1972), 50–71.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    F. R. Giles andW. R. Pulleyblank, Total dual integrality and integer polyhedra,Lin. Algebra and Appl. 25 (1979), 191–196.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    H. Gröflin,On a general antiblocking relation and some integer polyhedra, Habilitationsschrift ETH (Zürich, April 1982).

  8. [8]

    H. Gröflin, On node constraint networks,IFOR Technical Report, ETH (Zürich, December 1982).

  9. [9]

    H. Gröflin, On Switching paths polyhedra,IFOR Technical Report, ETH (Zürich, May 1982).

  10. [10]

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

    MATH  Google Scholar 

  11. [12]

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

    MATH  Article  MathSciNet  Google Scholar 

  12. [13]

    A. J. Hoffman, On lattice polyhedra II: construction and examples,IBM Research Rep. RC 6268, (1976).

  13. [14]

    A. J. Hoffman, On lattice polyhedra III: blockers and anti-blockers of lattice clutters,Math. Programming Study 9 (1978), 197–207.

    Google Scholar 

  14. [15]

    A. J. Hoffman andD. E. Schwartz, On lattice polyhedra, inCombinatorics: Proc. 5th Hung. Coll. on Comb. (A. Hajnal and V. T. Sós eds.) (North-Holland, Amsterdam, 1978), 593–598.

    Google Scholar 

  15. [16]

    E. L. Johnson, Support functions, blocking pairs and antiblocking pairs,Mathematical Programming Study 8 (1978), 167–196.

    Google Scholar 

  16. [17]

    A. Schrijver, On total dual integrality,Lin. Algebra and Appl. 38 (1981), 27–32.

    MATH  Article  MathSciNet  Google Scholar 

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

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AMS subject classification (1980)

  • 05 C 20
  • 05 C 38
  • 06 A 10
  • 52 A 25
  • 90 B 10
  • 90 C 05