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Submodular linear programs on forests

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Abstract

A general linear programming model for an order-theoretic analysis of both Edmonds' greedy algorithm for matroids and the NW-corner rule for transportation problems with Monge costs is introduced. This approach includes the model of Queyranne, Spieksma and Tardella (1993) as a special case. We solve the problem by optimal greedy algorithms for rooted forests as underlying structures. Other solvable cases are also discussed.

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References

  1. W.W. Bein, P. Brucker, J.K. Park and P.K. Pathak, “A Monge property for thed-dimensional transportation problem,”Discrete Applied Mathematics 58 (1995) 97–109.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Birkhoff,Lattice Theory, American Mathematical Society Colloquium Publications, Vol. 25 (American Mathematical Society, Providence, RI, 1967).

    MATH  Google Scholar 

  3. R.E. Burkard, B. Klinz and R. Rudolf, “Perspectives of Monge properties in optimization,”Discrete Applied Mathematics, to appear.

  4. U. Derigs, O. Goecke and R. Schrader, “Monge sequences and a simple assignment algorithm,”Discrete Applied Mathematics 15 (1986) 241–248.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Edmonds, “Submodular functions, matroids and certain polyhedra,” in: R. Guy et al., eds.,Combinatorial Structures and their Applications (Gordon and Breach, New York, 1970) pp. 69–87.

    Google Scholar 

  6. J. Edmonds and R. Giles, “A min-max relation for submodular functions on graphs,” in: P.L. Hammer, E.L. Johnson, B.H. Korte and G.L. Nemhauser, eds.,Studies in Integer Programming, Annals of Discrete Mathematics, Vol. 1 (North-Holland, Amsterdam 1977) pp. 185–204.

    Google Scholar 

  7. U. Faigle, “Matroids in combinatorial optimization,” in: N. White, ed.,Combinatorial Geometries, Encyclopedia of Mathematics and its Applications, Vol. 29 (Cambridge University Press, Cambridge, 1987) pp. 161–210.

    Google Scholar 

  8. A. Frank and E. Tardos, “Generalized polymatroids and submodular flows,”Mathematical Programming 42 (1988) 489–563.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. FujishigeSubmodular Functions and Optimization. Annals of Discrete Mathematics, Vol. 47 (North-Holland, Amsterdam, 1990).

    Google Scholar 

  10. A.J. Hoffman, “On simple linear programming problems,” in: V. Klee, ed.,Convexity, Proceedings of Symposia in Pure Mathematics, Vol. 7 (American Mathematical Society, Providence, RI, 1963) pp. 317–327.

    Google Scholar 

  11. M. Queyranne, F. Spieksma and F. Tardella, “A general class of greedily solvable linear program,” in: U. Faigle and C. Hoede, eds.,3rd Twente Workshop on Graphs and Combinatorial Optimization, Memorandum No. 1132 (Department of Applied Mathematics, University of Twente, Enschede, 1993).

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Ulrich Faigle & Walter Kern

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  1. Ulrich Faigle
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  2. Walter Kern
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Correspondence to Ulrich Faigle.

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Faigle, U., Kern, W. Submodular linear programs on forests. Mathematical Programming 72, 195–206 (1996). https://doi.org/10.1007/BF02592089

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  • DOI: https://doi.org/10.1007/BF02592089

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