Mathematical Programming

, Volume 98, Issue 1–3, pp 177–199 | Cite as

Facets of the independent set polytope

  • Todd Easton
  • Kevin Hooker
  • Eva K. LeeEmail author


Theoretical results pertaining to the independent set polytope P ISP =conv{x{0,1} n :Axb} are presented. A conflict hypergraph is constructed based on the set of dependent sets which facilitates the examination of the facial structure of P ISP . Necessary and sufficient conditions are provided for every nontrivial 0-1 facet-defining inequalities of P ISP in terms of hypercliques. The relationship of hypercliques and some classes of knapsack facet-defining inequalities are briefly discussed. The notion of lifting is extended to the conflict hypergraph setting to obtain strong valid inequalities, and back-lifting is introduced to strengthen cut coefficients. Preliminary computational results are presented to illustrate the usefulness of the theoretical findings.


independent set polytope conflict hypergraph hyperclique facet integer programming graph theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Atamturk, A., Nemhauser, G.L., Savelsbergh, M.W.P.: The Mixed Vertex Packing Problem. Mathematical Programming 89, 35–53 (2000)Google Scholar
  2. 2.
    Balas, E.: Facets of the Knapsack Polytope. Mathematical Programming 8, 146–164 (1975)Google Scholar
  3. 3.
    Balas, E.: Facets of One-Dimensional and Multi-Dimensional Knapsack Polytopes. Symposia Mathematica. XIX, Academic Press, 11-34 (1976)Google Scholar
  4. 4.
    Balas, E., Zemel, E.: Facets of the Knapsack Polytope from Minimal Covers. SIAM Journal on Applied Mathematics 34, 119–148 (1978)Google Scholar
  5. 5.
    Beasley, J.E.: OR-Library: distributing test problems by electronic mail. Journal of the Operational Research Society 41(11), 1069–1072 (1990)Google Scholar
  6. 6.
    Berge, C.: Graphs and Hypergraphs. North-Holland Publishing Company, translated by Minieka, E. 1973Google Scholar
  7. 7.
    Bixby, R.E., Lee, E.K.: Solving a Truck Dispatching Scheduling Problem Using Branch and Cut. Operations Research 46(3), 355–367 (1994)Google Scholar
  8. 8.
    Borndörfer, R.: Aspects of Set Packing, Partitioning, and Covering. Ph.D. thesis, Technischen Universtät Berlin, Berlin Germany. 1997Google Scholar
  9. 9.
    Chu, P.C., Beasley, J.E.: A Genetic Algorithm for the Multidimensional Knapsack Problem. Journal of Heuristics 4, 63–86 (1998)Google Scholar
  10. 10.
    Chvaal, V.: Edmonds Polytopes and Weakly a Hierarchy of Combinatorial problems. Discrete Mathematics 4, 305–337 (1973)Google Scholar
  11. 11.
    Cornuéjols G., Sassano, A.: On the 0,1 Facets of the Set Covering Polytope. Mathematical Programming 43, 45–55 (1989)Google Scholar
  12. 12.
    Easton, T., Hooker, K, Lee, E.K.: On Solving the Multiple Knapsack Problem – An Aggregate Branching and Hyperclique Cut Approach. working paper. 2002Google Scholar
  13. 13.
    Euler, R., Junger, M., Reinelt, G.: Generalizations of Cliques, Odd Cycles and Anticycles and their relation to Independence System Polyhedra. Mathematics of Operations Research 12(3), 451-462 (1987)Google Scholar
  14. 14.
    Gomory, R.E.: Some Polyhedra related to Corner Problems. Linear Algebra and its Applications 2, 451–588 (1969)Google Scholar
  15. 15.
    Gomory, R.E.: Outline of an Algorithm for Integer Solutions to Linear Programs. Bulletin of the American Mathematical Society 64, 275–278 (1958)Google Scholar
  16. 16.
    Hammer, P.L., Johnson, E.L., Peled, U.N.: Facets of Regular 0-1 Polytopes. Mathematical Programming 8, 179–206 (1975)Google Scholar
  17. 17.
    ILOG CPLEX 7.5 User's Manual, 2001Google Scholar
  18. 18.
    Karp, R.M.: Reducibility among Combinatorial Problems. In: R. E. Miller and J. W. Thatcher (eds.), Complexity of Computer Computations, Plenum Press, New York 1972, pp. 85–103Google Scholar
  19. 19.
    Lee, E.K.: Solving a Truck Dispatching Scheduling Problem Using Branch-and-Cut. Ph.D. thesis, Computational and Applied Mathematics, Rice University, Houston USA. 1993Google Scholar
  20. 20.
    Martin, A., Weismantel, R.: The Intersection of Knapsack Polyhedra and Extensions. Integer Programming and Combinatorial Optimization (Houston, TX), Lecture Notes in Computer Science, Springer, Berlin, 1998, pp. 243–256Google Scholar
  21. 21.
    Nemhauser, G.L., Trotter, L.: Properties of Vertex Packing and Independence System Polyhedra. Mathematical Programming 6, 48–61 (1974)Google Scholar
  22. 22.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley and SONS, New York. 1988Google Scholar
  23. 23.
    Padberg, M.: On the Facial Structure of Set Packing Polyhedra. Mathematical Programming 5, 199–215 (1973)Google Scholar
  24. 24.
    Padberg, M.: (1,k)-Configurations and Facets for Packing Problems. Mathematical Programming 18, 94–99 (1980)Google Scholar
  25. 25.
    Pulleyblank, W.R., Edmonds, J.: Facets of 1-Matching Polyhedra. Hypergraph Seminar. C. Berge and D. Ray-Chaudhuri, Eds., Springer 214–242 (1975)Google Scholar
  26. 26.
    Welsh, D.: Matroid Theory. Academic Press, (1976)Google Scholar
  27. 27.
    Wolsey, L.A.: Faces for a Linear Inequality in 0-1 Variables. Mathematical Programming 8, 165–178 (1975)Google Scholar
  28. 28.
    Wolsey, L.A.: Facets and Strong Valid Inequalities for Integer Programs. Operations Research 24, 367–372 (1975b)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.School of Industrial and Manufacturing Systems EngineeringKansas State UniversityManhattanKansas
  2. 2.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaGeorgia
  3. 3.Department of Radiation OncologyEmory University School of MedicineAtlantaGeorgia

Personalised recommendations