Abstract
We say that a polyhedronP satisfies weak integral decomposition if whenever an integral vector is the sum ofk vectors inP it is also the sum ofk integral vectors inP. This property is related to rounding results for packing and covering problems. We study the property and two related properties, and give results concerning integral polymatroids, totally unimodular matrices and network flows, pairs of strongly-base-orderable matroids, and branchings in directed graphs.
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References
Z. Baranyai, “The edge-coloring of complete hypergraphs, I”,Journal of Combinatorial Theory B26 (1979) 276–294.
S. Baum and L. E. Trotter, “Integer rounding and polyhedral decomposition for totally unimodular systems”, in: R. Henn, B. Korte and W. Oettli, eds,Arbeitstagung über Operations Research und Optimierung (Springer, Berlin, 1978) pp. 15–23.
S. Baum and L.E. Trotter, “Finite checkability for integer rounding properties in combinatorial programming problems”,Mathematical Programming (to appear).
S. Baum and L.E. Trotter, “Integer rounding for polymatroid and branching optimisation problems”,SIAM Journal on Algebraic and Discrete Methods (to appear).
C. Berge,Graphs and hypergraphs (North-Holland, Amsterdam, 1973).
R.A. Brualdi, “Induced matroids”,Proceedings of the American Mathematical Society 29 (1971) 213–221.
J. Davies and C. McDiarmid, “Disjoint common transversals and exchange structures”,Journal of the London Mathematical Society 14(2) (1976) 55–62.
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) pp. 69–87.
J. Edmonds, “Edge-disjoint branchings”, in: R. Rustin, ed.,Combinatorial algorithms (Algorithmics Press, New York, 1972) pp. 91–96.
S. Even, A. Itai and A. Shamir, “On the complexity of timetable and multicommodity flow problems”,SIAM Journal on Computing 5 (1976) 691–703.
D.R. Fulkerson, “Blocking and anti-blocking pairs of polyhedra”,Mathematical Programming 1 (1971) 168–194.
D.R. Fulkerson, “Packing rooted directed cuts in a weighted directed graph”,Mathematical Programming 6 (1974) 1–13.
D.R. Fulkerson and D.B. Weinberger, “Blocking pairs of polyhedra arising from network flows”,Journal of Combinatorial Theory 18 (1975) 265–283.
R.S. Garfinkel and G.L. Nemhauser,Integer programming (Wiley, New York, 1972).
F.R. Giles, “Submodular functions, graphs and integer polyhedra”, Ph.D. thesis, University of Waterloo, Waterloo, Ontario, Canada (1975).
A.J. Hoffman and J.G. Kruskal, “Integral boundary points of convex polyhedra”, in: H.W. Kuhn and A.W. Tucker, eds.,Linear inequalities and related systems, Annals of Mathematics Studies 38 (Princeton University Press, Princeton, NJ, 1956) pp. 223–246.
E.L. Lawler,Combinatorial optimization: Networks and matroids (Holt. Rinehart and Winston. New York, 1976).
M.D. McDaniel, “Network models for linear programming problems with integer rounding properties”, M.S. thesis, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY (1981).
C.J.H. McDiarmid, “Rado's theorem for polymatroids”,Mathematical Proceedings of the Cambridge Philosophical Society 78 (1975) 263–281.
C.J.H. McDiarmid, “On pairs of strongly-base-orderable matroids”, Tech. Rep. No. 283, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY (1976).
C.J.H. McDiarmid, “Blocking, anti-blocking and pairs of matroids and polymatroids”,Journal of Combinatorial Theory B25 (1978) 313–325.
J.B. Orlin, “A polynomial algorithm for integer programming covering problems satisfying the integer round-up property”, Sloan School of Management Tech. Rept., MIT, Cambridge, MA (1980).
A. Schrijver, “Matroids and linking systems”,Mathematical Centre Tracts 88, Mathematisch Centrum, Amsterdam (1977).
L.E. Trotter and D.B. Weinberger, “Symmetric blocking and anti-blocking relations for generalized circulations”,Mathematical Programming Study 8 (1978) 141–158.
D.B. Weinberger, “Transversal matroid intersections and related packings”,Mathematical Programming 11 (1976) 164–176.
D.B. Weinberger, “Network flows, minimum coverings, and the four-color conjecture”,Operations Research 24 (1976) 272–290.
D.J.A. Welsh,Matroid theory (Academic Press, London, 1976).
D. de Werra, “Balanced schedules”,INFOR 9 (1971) 230–237.
D. de Werra, “Equitable colorations of graphs”,Revue Française d'Informatique et de Recherche Opérationnelle R-3 (1971) 3–8.
D. de Werra, “On some characterisations of totally unimodular matrices”,Mathematical Programming 20 (1981) 14–21.
D. Woodwall, “The induction of matroids by graphs”,Journal of the London Mathematical Society 10(2) (1975) 27–35.
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McDiarmid, C. Integral decomposition in polyhedra. Mathematical Programming 25, 183–198 (1983). https://doi.org/10.1007/BF02591770
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DOI: https://doi.org/10.1007/BF02591770