Abstract
Edmonds and Giles introduced the class of box totally dual integral polyhedra as a generalization of submodular flow polyhedra. In this paper a geometric characterization of these polyhedra is given. This geometric result is used to show that each TDI defining system for a box TDI polyhedron is in fact a box TDI system, that the class of box TDI polyhedra is in co-NP and is closed under taking projections and dominants, that the class of box perfect graphs is in co-NP, and a result of Edmonds and Giles which is related to the facets of box TDI polyhdera.
Similar content being viewed by others
References
A. Bachem and M. Grötschel, “New aspects of polyhedral theory“, in: B. Korte, ed.,Modern applied mathematics — Optimization and operations research (North-Holland, Amsterdam, 1982) 51–106.
E. Balas and W.R. Pulleyblank, “The perfectly matchable subgraph polytope of a bipartite graph“,Networks 13 (1983) 495–516.
C. Berge,Graphs and hypergraphs (North-Holland, Amsterdam, 1973).
C. Berge and V. Chvátal,Topics on perfect graphs (North-Holland, Amsterdam, 1984).
K. Cameron, “Polyhedral and algorithmic ramifications of antichains”, Ph.D. Thesis, University of Waterloo, 1982.
R. Chandrasekaran, “Polynomial algorithms for totally dual integral systems and extensions“,Annals of Discrete Mathematics 11 (1981) 39–51.
W. Cook, L. Lovász and A. Schrijver, “A polynomial-time test for total dual integrality in fixed dimension“,Mathematical Programming Study 22 (1984) 64–69.
W.H. Cunningham, “An unbounded matroid intersection polyhedron“, Linear Algebra and its Appliications 16 (1977) 209–215.
W.H. Cunningham and J. Green-Krotki, to appear.
J. Edmonds, “Submodular functions, matroids, and certain polyhedra“, in: R. Guy, H. Hanani, N. Sauer and J. Schonheim, eds.,Combinatorial structures and their applications (Gordon and Breach, New York, 1970) 69–87.
J. Edmonds and R. Giles, “A min-max relation for submodular functions on graphs“,Annals of Discrete Mathematics 1 (1977) 185–204.
J. Edmonds and R. Giles, “Box total dual integrality”, unpublished paper, 1980.
J. Edmonds and R. Giles, “Total dual integrality of linear inequality systems“, in: W.R. Pulleyblank, ed.,Progress in combinatorial optimization (Academic Press, Toronto, 1984) 117–129.
A. Frank, “Kernel systems of directed graphs“,Acta Scientiarum Mathematicarum (Szeged) 41 (1979) 63–76.
A. Frank, “Generalized polymatroids”, Report 81206-OR, Institut für Ökonometrie und Operations Research, Bonn, F.R. Germany, 1981.
A. Frank and E. Tardos, “Generalized polymatroids and submodular flows”, in preparation.
D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra“,Mathematical Programming 1 (1971) 168–194.
M.R. Garey and D.S. Johnson,Computers and intractability: A Guide to the theory of NP-completeness (W.H. Freeman and Company, San Francisco, 1979).
J. von zur Gathen and M. Sieveking, “A bound on solutions of linear integer equalities and inequalities“,Proceedings of the American Mathematical Society 72 (1978) 155–158.
F.R. Giles and W.R. Pulleyblank, “Total dual integrality and integer polyhedra“, Linear Algebra and its Applications 25 (1979) 191–196.
C. Greene and D. Kleitman, “The structure of Spernerk-families“,Journal of Combinatorial Theory Series A 20 (1976) 41–68.
H. Gröflin and A.J. Hoffman, “Lattice polyhedra II: Generalization, constructions and examples“,Annals of Discrete Mathematics 15 (1982) 189–203.
M. Grötschel, L. Lovász and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization“,Combinatorica 1 (1981) 169–197.
M. Grötschel, L. Lovász and A. Schrijver, “Polynomial algorithms for perfect graphs“, in: C. Berge and V. Chvátal, eds.,Topics on perfect graphs (North-Holland, Amsterdam, 1984) 325–356.
M. Grötschel, L. Lovász and A. Schrijver, “Relaxations of vertex packing”, Report 84341-OR, Institut für Ökonometrie und Operations Research, Bonn, W. Germany, 1984.
M. Grötschel, L. Lovász and A. Schrijver,The ellipsoid method and combinatorial optimization (Springer-Verlag), to appear.
A.J. Hoffman, “A generalization of max flow-min cut“,Mathematical Programming 6 (1974) 352–359.
A.J. Hoffman and D.E. Schwartz, “On lattice polyhedra“, in: A. Hajnal and V.T. Sós, eds.,Combinatorics (North-Holland, Amsterdam, 1978) 593–598.
L.G. Khachiyan, “A polynomial algorithm in linear programming“,Soviet Mathematics Doklady 20 (1979) 191–194.
L. Lovász, “Normal hypergraphs and the perfect graph conjecture“,Discrete Mathematics 2 (1972) 253–267.
L. Lovász, “Perfect graphs“, in: L.W. Beineke and R.J. Wilson, eds.,Selected topics in graph theory 2 (Academic Press, London, 1983) 55–87.
C. McDiarmid, “Blocking, antiblocking, and pairs of matroids and polymatroids“,Journal of Combinatorial Theory Series B 25 (1978) 313–325.
W.R. Pulleyblank, “Polyhedral combinatorics“, in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming—The state of the art (Springer-Verlag, Heidelberg, 1983) 312–345.
R.T. Rockafeller,Convex analysis (Princeton University Press, Princeton, 1970).
A. Schrijver, “On total dual integrality“,Linear Algebra and its Applications 38 (1981) 27–32.
A. Schrijver, “Proving total dual integrality with cross-free families—a general framework“,Mathematical Programming 29 (1984) 15–27.
A. Schrijver, “Total dual integrality from directed graphs, crossing families, and sub- and supermodular functions“, in: W.R. Pulleyblank, ed.,Progress in combinatorial optimization (Academic Press, Toronto, 1984) 315–361.
A. Schrijver,Theory of linear and integer programming (Wiley) to appear.
Author information
Authors and Affiliations
Additional information
Supported by a grant from the Alexander von Humboldt-Stiftung.
Rights and permissions
About this article
Cite this article
Cook, W. On box totally dual integral polyhedra. Mathematical Programming 34, 48–61 (1986). https://doi.org/10.1007/BF01582162
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582162