Mathematical Programming

, Volume 62, Issue 1–3, pp 1–14 | Cite as

Series parallel composition of greedy linear programming problems

  • Wolfgang W. Bein
  • Peter Brucker
  • Alan J. Hoffman


We study the concept of series and parallel composition of linear programming problems and show that greedy properties are inherited by such compositions. Our results are inspired by earlier work on compositions of flow problems. We make use of certain Monge properties as well as convexity properties which support the greedy method in other contexts.

Key words

Greedy algorithm Monge arrays series parallel graphs linear programming network flow transportation problem integrality convexity 


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  1. [1]
    Y.P. Aneja, R. Chandrasekaran and K.P.K. Nair, “Classes of linear programs with integral optimal solutions,”Mathematical Programming Study 25 (1985) 225–237.Google Scholar
  2. [2]
    W.W. Bein, “Netflows, polymatroids, and greedy structures,” Ph.D. Thesis, Universität Osnabrück (Osnabrück, Germany, 1986).Google Scholar
  3. [3]
    W.W. Bein and P. Brucker, “Greedy concepts for network flow problems,”Discrete Applied Mathematics 15 (1986) 135–144.Google Scholar
  4. [4]
    W.W. Bein, P. Brucker and A. Tamir, “Minimum cost flow algorithms for series parallel networks,”Discrete Applied Mathematics 10 (1985) 117–124.Google Scholar
  5. [5]
    P.C. Gilmore, E.L. Lawler and D.B. Shmoys, “Well-solved special cases,” in: E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, eds.,The Travelling Salesman Problem — A Guided Tour of Combinatorial Optimization (Wiley, New York, 1985) pp. 87–143.Google Scholar
  6. [6]
    G.H. Hardy, J.E. Littlewood and G. Polya,Inequalities (Cambridge University Press, Cambridge, England, 1934).Google Scholar
  7. [7]
    A.J. Hoffman, “On simple linear programming problems,” in: V.L. Klee, ed.,Convexity, Proceedings of Symposia in Pure Mathematics, Vol. 7 (American Mathematical Society, Providence, RI, 1963) pp. 317–327.Google Scholar
  8. [8]
    A.J. Hoffman, “On greedy algorithms that succeed,” in: I. Anderson, ed.,Surveys in Combinatorics 1985. London Mathematical Society Lecture Note Series No. 103 (Cambridge University Press, Cambridge, England, 1985) pp. 97–112.Google Scholar
  9. [9]
    A.J. Hoffman, “On greedy algorithms for series parallel graphs,”Mathematical Programming 40 (1988) 197–204.Google Scholar
  10. [10]
    J. Kahn, oral communication (1988).Google Scholar
  11. [11]
    G. Monge, “Mémoire sur la théorie des déblai et des remblai,”Histoire de l'Academie Royale des Sciences (année 1781) (Paris, 1784) pp. 666–704.Google Scholar
  12. [12]
    C.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization (Prentice-Hall, Englewood Cliffs, NJ, 1982).Google Scholar
  13. [13]
    J. Valdes, “Parsing flowcharts and series-parallel graphs,” Ph.D. Thesis, Stanford University (Stanford, CA, 1978).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Wolfgang W. Bein
    • 1
  • Peter Brucker
    • 2
  • Alan J. Hoffman
    • 3
  1. 1.American Airlines Decision TechnologiesDallas/Fort Worth AirportUSA
  2. 2.Fachbereich Mathematik/InformatikUniversität OsnabrückGermany
  3. 3.Department of Mathematical ScienceIBM Thomas J. Watson Research CenterYorktown HeightsUSA

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