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Mathematical Programming Computation

, Volume 5, Issue 1, pp 57–73 | Cite as

Implementation of a unimodularity test

  • Matthias Walter
  • Klaus Truemper
Full Length Paper

Abstract

This paper describes implementation and computational results of a polynomial test of total unimodularity. The test is a simplified version of a prior method. The program also decides two related unimodularity properties. The software is available free of charge in source code form under the Boost Software License.

Keywords

Unimodularity Total unimodularity Polynomial test 

Mathematics Subject Classification

05-04 combinatorics - explicit machine computation and programming 

References

  1. 1.
    Bixby, R.E., Cunningham, W.H.: Converting linear programs to network problems. Math. Oper. Res. 5, 321–357 (1980)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bixby, R.E., Cunningham, W.H., Rajan, R.: A decomposition algorithm for matroids. Rice University, Technical report (1986)Google Scholar
  3. 3.
    Boost Software License http://www.boost.org/LICENSE_1_0.txt
  4. 4.
    Camion, P.: Matrices Totalement Unimodulaire et Problèmes Combinatoires. PhD thesis, Université Libre de Bruxelles, Bruxelles (1963)Google Scholar
  5. 5.
    Cunningham, W.H., Edmonds, J.: Decomposition of linear systems, (unpublished). (1965)Google Scholar
  6. 6.
    Edmonds, J.: Minimum partition of a matroid into independet subsets. J. Res. Natl. Bur. Stand. (B) 69, 67–72 (1965)MathSciNetMATHGoogle Scholar
  7. 7.
    Fujishige, S.: An efficient \(pq\)-graph algorithm for solving the graph-realization problem. J. Comput. Syst. Sci. 21, 63–86 (1980)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ghouila-Houri, A.: Caracterisation des matrices totalement unimodulaires. C.R. Acad. Sci. Paris 254, 1192–1194 (1962)MathSciNetMATHGoogle Scholar
  9. 9.
    Gilbert, E.N.: Random graphs. Ann. Math. Stat. 30, 1141–1144 (1959)MATHCrossRefGoogle Scholar
  10. 10.
    Hoffman, A.J., Kruskal, J.B.: Integral boundary points of convex polyhedra. Ann. Math. Stud. 38, 223–246 (1956)MathSciNetMATHGoogle Scholar
  11. 11.
    Hoffman, A.J., Oppenheim, R.: Local unimodularity in the matching polytope. Ann. Discret. Math. 2, 201–209 (1978)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Seymour, P.D.: Decomposition of regular matroids. J. Comb. Theory Ser. B 28, 305–359 (1980)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Smith, H.J.S.: On systems of linear indeterminate equations and congruences. Philos. Trans. R. Soc. Lond. 151, 293–326 (1861–1862)Google Scholar
  14. 14.
  15. 15.
    Truemper, K.: Algebraic characterizations of unimodular matrices. SIAM J. Appl. Math. 35, 328–332 (1978)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Truemper, K.: Complement total unimodularity. Linear Algebra Appl. 30, 77–92 (1980)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Truemper, K.: A decomposition theory for matroids. V. Testing of matrix total unimodularity. J. Comb. Theory Ser. B 49, 241–281 (1990)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Truemper, K.: A decomposition theory for matroids. VII. Analysis of minimal violation matrices. J. Comb. Theory Ser. B 55, 302–335 (1992)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Truemper, K.: Matroid Decomposition (revised edn). Leibniz, Plano, TX (1998)Google Scholar
  20. 20.
    Truemper, K., Chandrasekaran, R.: Local unimodularity of matrix–vector pairs. Linear Algebra Appl. 22, 65–78 (1978)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Tutte, W.T.: A homotopy theorem for matroids I, II. Trans. Am. Math. Soc. 88, 527–552 (1958)MathSciNetGoogle Scholar
  22. 22.
    Whitney, H.: 2-isomorphic graphs. Am. J. Math. 55, 245–254 (1933)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Institute of Mathematical OptimizationUniversity of Magdeburg “Otto von Guericke”MagdeburgGermany
  2. 2.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

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