Mathematical Programming

, Volume 8, Issue 1, pp 232–248 | Cite as

Vertex packings: Structural properties and algorithms

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


We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.


Optimality Condition Feasible Solution Integer Programming Computational Experience Programming Formulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    E. Balas and H. Samuelsson, “Finding a minimum node cover in an arbitrary graph”, Management Sciences Research Rept. No. 325, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, Pa. (November 1973).Google Scholar
  2. [2]
    M.L. Balinski, “On maximum matching, minimum covering and their connections”, in: H.W. Kuhn, ed.,Proceedings of the Princeton symposium on mathematical programming, (Princeton University Press, Princeton, N.J., 1970) pp. 303–312.Google Scholar
  3. [3]
    C. Berge,The theory of graphs and its applications (Methuen, London, 1962).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]
    V. Chvátal, “Edmonds polytopes and a hierarchy of combinatorial problems”,Discrete Mathematics 5 (1973) 305–337.Google Scholar
  6. [6]
    J. Edmonds, “Covers and packings in a family of sets”,Bulletin of the American Mathematical Society 68 (1962) 494–499.Google Scholar
  7. [7]
    J. Edmonds, “Paths, trees, and flowers”,Canadian Journal of Mathematics 17 (1965) 449–467.Google Scholar
  8. [8]
    L.R. Ford, Jr. and D.R. Fulkerson,Flows in networks (Princeton University Press, Princeton, N.J., 1962).Google Scholar
  9. [9]
    D.R. Fulkerson, “Anti-blocking polyhedra”,Journal of Combinatorial Theory 12 (1972) 50–71.Google Scholar
  10. [10]
    F. Gavril, “Algorithms for minimum coloring, maximum clique, minimum covering by cliques and maximum independent set of a chordal graph”,SIAM Journal of Computing 1 (1972) 180–187.Google Scholar
  11. [11]
    A.M. Geoffrion, “An improved implicit enumeration approach for integer programming”,Operations Research 17 (1969) 437–454.Google Scholar
  12. [12]
    R.M. Karp, “Reducibility among combinatorial problems”, in: R.E. Miller, J.W. Thatcher and J.D. Bohlinger, eds.,Complexity of computer computations (Plenum Press, New York, 1972) pp. 85–103.Google Scholar
  13. [13]
    C.E. Lemke, H.M. Salkin and K. Spielberg, “Set covering by single branch enumeration with linear programming subproblems”,Operations Research 19 (1971) 998–1022.Google Scholar
  14. [14]
    G.L. Nemhauser and L.E. Trotter, Jr., “Properties of vertex packing and independence system polyhedra”,Mathematical Programming 6 (1974) 48–61.Google Scholar
  15. [15]
    M. Padberg, “On the facial structure of set packing polyhedra”,Mathematical programming 5 (1973) 199–216.Google Scholar
  16. [16]
    R. Tarjan, “Finding a maximum clique”, Tech. Rept. 72-123, Dept. of Computer Science, Cornell University, Ithaca, N.Y. (March 1972).Google Scholar
  17. [17]
    L.E. Trotter, Jr., “Solution characteristics and algorithms for the vertex packing problem”, Tech. Rept. No. 168, Dept. of Operations Research, Cornell University, Ithaca, N.Y. (January 1973).Google Scholar
  18. [18]
    L.E. Trotter, Jr., “A class of facet producing graphs for vertex packing polyhedra”, Research Rept. No. 78, Dept. of Administrative Sciences, Yale University, New Haven, Conn. (February 1974).Google Scholar

Copyright information

© The Mathematical Programming Society 1975

Authors and Affiliations

  • G. L. Nemhauser
    • 1
  • L. E. TrotterJr.
    • 2
  1. 1.Cornell UniversityIthacaUSA
  2. 2.Yale UniversityNew HavenUSA

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