Abstract
We study the max cut problem in graphs not contractible toK 5, and optimum perfect matchings in planar graphs. We prove that both problems can be formulated as polynomial size linear programs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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).
F. Barahona, “The max cut problem on graphs not contractible toK 5,”Operations Research Letters 2 (1983a) 107–111.
F. Barahona, “On some weakly bipartite graphs,”Operations Research Letters 2 (1983b) 239–242.
F. Barahona, “Reducing matching to polynomial size linear programming,” (1988), to appear in:SIAM Journal on Optimization.
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.
F. Barahona and A.R. Mahjoub “On the Cut Polytope,”Mathematical Programming 36 (1986a) 157–173.
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).
F. Barahona and A.R. Mahjoub, “Compositions of graphs and polydedra II: Stable Sets,” (19887), to appear in:SIAM Journal on Discrete Mathematics.
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.
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.
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.
J. Edmonds and E.L. Johnson, “Matching, Euler tours and the Chinese Postman,”Mathematical Programming 5 (1973) 88–124.
J. Fonlupt, A.R. Mahjoub and J.P. Uhry, “Compositions ion the bipartite subgraph polytope,” (1984), to appear in:Discrete Mathematics.
D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra,”Mathematical Programming 1 (1971) 168–194.
T.C. Hu, “Multicommodity network flows,”Operations Research 11 (1963) 344–360.
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).
B. Rothschild and A. Whinston, “Feasibility of two-commodity network flows,”Operations Research 14 (1966) 1121–1129.
A. Schrijver,Theory of Linear and Integer Programming (Wiley, New York, 1986).
P.D. Seymour, “On odd cuts and planar multicommodity flows,”Proceedings of the London Mathematical Society 42 (1981a) 178–192.
P.D. Seymour, “Matroids and multicommodity flows,”European Journal of Combinatorics 2 (1981b) 257–290.
K. Wagner, “Beweis einer Abschwächung der Hadwiger-Vermutung,”Mathematische Annalen 153 (1964) 139–141.
K. Wagner,Graphentheorie (Hochschultaschenbücher-Verlag, Berlin, 1970).
R.T. Wong, “A dual ascent approach to Steiner tree problems in graphs,”Mathematical Programming 28 (1984) 271–287.
M. Yannakakis, “Expressing combinatorial optimization problems by linear programs,”Proceedings of the 29th IEEE Symposium on Foundations of Computer Science (1988) 223–228.
Author information
Authors and Affiliations
Additional information
Supported by the joint project “Combinatorial Optimization” of the Natural Sciences and Engineering Research Council of Canada and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).
Rights and permissions
About this article
Cite this article
Barahona, F. On cuts and matchings in planar graphs. Mathematical Programming 60, 53–68 (1993). https://doi.org/10.1007/BF01580600
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580600