Mathematical Programming

, Volume 6, Issue 1, pp 48–61 | Cite as

Properties of vertex packing and independence system polyhedra

  • G. L. Nemhauser
  • L. E. TrotterJr.


We consider two convex polyhedra related to the vertex packing problem for a finite, undirected, loopless graphG with no multiple edges. A characterization is given for the extreme points of the polyhedron\(\mathcal{L}_G = \{ x \in R^n :Ax \leqslant 1_m ,x \geqslant 0\} \), whereA is them × n edge-vertex incidence matrix ofG and 1m is anm-vector of ones. A general class of facets of
= convex hull{xRn:Ax≤1m,x binary} is described which subsumes a class examined by Padberg [13]. Some of the results for
are extended to a more general class of integer polyhedra obtained from independence systems.


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  1. [1]
    M.L. Balinski, “Integer programming: Methods, uses, computation”,Management Science 12 (1965) 253–313.Google Scholar
  2. [2]
    M.L. Balinski, “Establishing the matching polytope”,Journal of Combinatorial Theory 13 (1972) 1–13.Google Scholar
  3. [3]
    M.L. Balinski and K. Spielberg, “Methods of integer programming: Algebraic, combinatorial and enumerative”, in:Progress in operations research, Vol. III, Ed. J. Aronofsky (Wiley, New York, 1969) pp. 195–292.Google Scholar
  4. [4]
    V. Chvátal, “On certain polytopes associated with graphs”, Centre de Recherches Mathématiques-238, Université de Montréal (October 1972).Google Scholar
  5. [5]
    J. Edmonds, “Covers and packings in a family of sets”,Bulletin of the American Mathematical Society 68 (1962) 494–499.Google Scholar
  6. [6]
    J. Edmonds, “Maximum matching and a polyhedron with (0, 1)-vertices”,Journal of Research of the National Bureau of Standards 69B (1965) 125–130.Google Scholar
  7. [7]
    J. Edmonds, “Matroids and the greedy algorithm”,Mathematical Programming 1 (1971) 127–136.Google Scholar
  8. [8]
    D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra”,Mathematical Programming 1 (1971) 168–194.Google Scholar
  9. [9]
    D.R. Fulkerson, “Anti-blocking polyhedra”,Journal of Combinatorial Theory 12 (1972) 50–71.Google Scholar
  10. [10]
    R.S. Garfinkel and G.L. Nemhauser, “A survey of integer programming emphasizing computation and relations among models”, in:Mathematical Programming Eds. T.C. Hu and S.M. Robinson (Academic Press, New York, 1973) pp. 77–155.Google Scholar
  11. [11]
    R.E. Gomory, “Some polyhedra related to combinatorial problems”,Linear Algebra and Its Applications 2 (1969) 451–558.Google Scholar
  12. [12]
    L. Lovász, “Normal hypergraphs and the perfect graph conjecture”,Discrete Mathematics 2 (1972) 253–267.Google Scholar
  13. [13]
    M.W. Padberg, “On the facial structure of set packing polyhedra”,Mathematical Programming 5 (1973) 199–215.Google Scholar
  14. [14]
    L.E. Trotter, Jr., “Solution characteristics and algorithms for the vertex packing problem”, Technical Report No. 168. Department of Operations Research, Cornell University (January 1973).Google Scholar
  15. [15]
    L.E. Trotter, Jr., “A class of facet producing graphs for vertex packing polyhedra”, Technical Report No. 78, Department of Administrative Sciences, Yale University, (February 1974).Google Scholar

Copyright information

© The Mathematical Programming Society 1974

Authors and Affiliations

  • G. L. Nemhauser
    • 1
  • L. E. TrotterJr.
    • 1
  1. 1.Cornell UniversityIthacaUSA

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