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Mathematical Programming

, Volume 60, Issue 1–3, pp 53–68 | Cite as

On cuts and matchings in planar graphs

  • Francisco Barahona
Article

Abstract

We study the max cut problem in graphs not contractible toK5, and optimum perfect matchings in planar graphs. We prove that both problems can be formulated as polynomial size linear programs.

Key words

Cut polytope matching multicommodity flows 

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References

  1. M. Ball, W.G. Liu and W.R. Pulleyblank, (1987), “Two terminal Steiner tree polyhedra,” Report 87466-OR, Institut für Operations Research Universität Bonn (Bonn, 1987).Google Scholar
  2. F. Barahona, “The max cut problem on graphs not contractible toK 5,”Operations Research Letters 2 (1983a) 107–111.Google Scholar
  3. F. Barahona, “On some weakly bipartite graphs,”Operations Research Letters 2 (1983b) 239–242.Google Scholar
  4. F. Barahona, “Reducing matching to polynomial size linear programming,” (1988), to appear in:SIAM Journal on Optimization. Google Scholar
  5. F. Barahona, “Planar multicommodity flows, max cut and the Chinese Postman Problem,” in:Polyhedral Combinatorics, DIMACS Series on Discrete Mathematics and Theoretical Computer Science No. 1 (DIMACS, NJ, 1990) pp. 189–202.Google Scholar
  6. F. Barahona and A.R. Mahjoub “On the Cut Polytope,”Mathematical Programming 36 (1986a) 157–173.Google Scholar
  7. F. Barahona and A.R. Mahjoub, “Compositions of graphs and polyhedra I: Balanced and Acyclic induced subgraphs,” Research Report CORR 86-16, University of Waterloo (Waterloo, Ont., 1986b).Google Scholar
  8. F. Barahona and A.R. Mahjoub, “Compositions of graphs and polydedra II: Stable Sets,” (19887), to appear in:SIAM Journal on Discrete Mathematics. Google Scholar
  9. F. Barahona and A.R. Mahjoub, “Compositions of graphs and polyhedra III: Graphs with noW 4 minor,” (1989), to appear in:SIAM Journal on Discrete Mathematics. Google Scholar
  10. G. Cornuéjols, D. Naddef and W.R. Pulleyblank “The Traveling Salesman Problem in Graphs with 3-edge cutsets,”Journal of the Association for Computing Machinery 32 (1985) 383–410.Google Scholar
  11. J. Edmonds, “Maximum matching and a polyhedron with (0, 1)-vertices,”Journal of the Research of the National Bureau of Standards 69B (1965) 125–130.Google Scholar
  12. J. Edmonds and E.L. Johnson, “Matching, Euler tours and the Chinese Postman,”Mathematical Programming 5 (1973) 88–124.Google Scholar
  13. J. Fonlupt, A.R. Mahjoub and J.P. Uhry, “Compositions ion the bipartite subgraph polytope,” (1984), to appear in:Discrete Mathematics. Google Scholar
  14. D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra,”Mathematical Programming 1 (1971) 168–194.Google Scholar
  15. T.C. Hu, “Multicommodity network flows,”Operations Research 11 (1963) 344–360.Google Scholar
  16. N. Maculan, “A new linear programming formulation for the shortest s-directed spanning tree problem,” Technical report ES 54–85, Systems Engineering and Computer Science, COPPE, Federal University of Rio de Janeiro (Rio de Janeiro, 1985).Google Scholar
  17. B. Rothschild and A. Whinston, “Feasibility of two-commodity network flows,”Operations Research 14 (1966) 1121–1129.Google Scholar
  18. A. Schrijver,Theory of Linear and Integer Programming (Wiley, New York, 1986).Google Scholar
  19. P.D. Seymour, “On odd cuts and planar multicommodity flows,”Proceedings of the London Mathematical Society 42 (1981a) 178–192.Google Scholar
  20. P.D. Seymour, “Matroids and multicommodity flows,”European Journal of Combinatorics 2 (1981b) 257–290.Google Scholar
  21. K. Wagner, “Beweis einer Abschwächung der Hadwiger-Vermutung,”Mathematische Annalen 153 (1964) 139–141.Google Scholar
  22. K. Wagner,Graphentheorie (Hochschultaschenbücher-Verlag, Berlin, 1970).Google Scholar
  23. R.T. Wong, “A dual ascent approach to Steiner tree problems in graphs,”Mathematical Programming 28 (1984) 271–287.Google Scholar
  24. M. Yannakakis, “Expressing combinatorial optimization problems by linear programs,”Proceedings of the 29th IEEE Symposium on Foundations of Computer Science (1988) 223–228.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Francisco Barahona
    • 1
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooCanada

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