The Theorem of Minkowski for Polyhedral Monoids and Aggregated Linear Diophantine Systems

  • Achim Bachem
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 157)


We study polyhedral monoids of the form M = {xεZn / Ax ≤ 0} for (m, n) integer matrices with rank m and prove in an elementary and constructive way that M has a finite basis, i.e. every x#x03B5;M is the nonnegative integer linear combination of a finite set of vectors. We show that this theorem holds also for monoids M(N, B)={xɛZ + S / Nx + By=o, yɛZ>n}. We consider the aggregated system GNx+GBy=o where G is an (r,m) aggregation matrix and show how the cardinality of a span of M(GN,GB) and M(N,B) relate to each other. Moreover we show how the group order of the Gomory group derived from M(N,B) changes if we aggregate Nx+By=o to GNx+GBy=o.


Aggregation Matrix Group Order Integer Matrix Finite Basis Aggregate System 
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  1. 1.
    Bachern, A., “Reduction And Decompositions Of Integer Programs Over Cones”, Annals of Discrete Mathematics 1, 1–11 (1977)CrossRefGoogle Scholar
  2. 2.
    Fiorot, J.Ch., “Generation of all integer points for given sets of linear inequalities.”, Mathematical Programming 3, 276–295 (1972)CrossRefGoogle Scholar
  3. 3.
    Gomory, R.E., “On The Relation Between Integer And Noninteger Solution To Linear Programs”, Proc. nat. Acad. Sci. USA, Vol. 53, p. 260–265, 1965CrossRefGoogle Scholar
  4. 4.
    Graver, J.E., “On the foundations of linear and integer programming I”, Mathematical Programming 9, 207–226 (1975)CrossRefGoogle Scholar
  5. 5.
    Hilbert, D., “Über die Theorie der algebraischen Formen”, Mathematische Annalen 36, 473–534 (1890)CrossRefGoogle Scholar
  6. 6.
    Jeroslow, R.G., “Some Structure And Basis Theorems For Integral Monoids”, Management Sciences Research Report No. 367, Graduate School of Industrial Administration, Carnegie-Mellon-University, 1975Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  • Achim Bachem
    • 1
  1. 1.Institut für Ökonometrie und Operations ResearchUniversität BonnBonnGermany

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