Abstract
Recent results show that edge-directions of polyhedra play an important role in (combinatorial) optimization; in particular, a d-dimensional polyhedron with |D| distinct edge-directions has at most O(|D|d-1) vertices. Here, we obtain a characterization of the directions of edges that are adjacent to a given vertex of a standard polyhedron of the form P = {x : Ax = b, l⩽ x⩽ u, tightening a standard necessary condition which asserts that such directions must be minimal support solutions of the homogenous equation Ax = 0 which are feasible at the given vertex. We specialize the characterization for polyhedra that correspond to network flows, obtaining a graph characterization of circuits which correspond to edge-directions. Applications to partitioning polyhedra are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Ben-Israel (1988) ArticleTitleCanonical bases in linear programming Linear Algebra and Its Applications 102 95–119 Occurrence Handle10.1016/0024-3795(88)90322-9
Dantzig, G. (1951), Maximization of a linear function of variables subject to linear inequalities. In: Koopman, T.C. (ed.), Activity Analysis of Production and Allocation, Chapter XXI, Cowles Commission Monograph 13, Wiley, New York.
P. Gritzmann B. Sturmfels (1993) ArticleTitleMinkowski addition of polytopes: computational complexity and applications to Gröbner bases SIAM Journal on Discrete Mathematics 6 246–269 Occurrence Handle10.1137/0406019
F.K. Hwang U.G. Rothblum (1996) ArticleTitleDirectional-quasi-convexity, asymmetric Schur convexity and optimality of consecutive partitions Mathematics of Operations Research 21 540–554
S. Onn U.G. Rothblum (2004) ArticleTitleConvex combinatorial optimization Discrete and Computational Geometry 32 549–566 Occurrence Handle10.1007/s00454-004-1138-y
T. Rockafellar (1970) Convex Analysis Princeton University Press Princeton, NJ
Rockafellar, T. (1984), Network Flows and Monotropic Optimization, Wiley Interscience.
Rothblum, U.G. and Tangir, Y. (2004), The Partition Bargaining Problem, Discrete Applied Math., to appear.
A. Schrijver (1986) Theory of Linear and Integer Programming John Wiley and Sons New York
Schultz, A.S., Weisamntel, R. and Ziegler, G. (1995), (0–1)-integer programming: Optimization and augmentation are equivalent. In: Proceedings of the Third Annual European Symposium on Algorithms, Lecture Notes in Computer Science 979, Springer, Berlin, Germany, pp. 473–483.
Tardella Fabio (1990) ArticleTitleOn the equivalence between some discrete and continuous optimization problems Annals of Operations Research 2 291–300
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of these authors was supported by a grant from ISF – the Israel Science Foundation, by a VPR grant at the Technion, and by the Fund for the Promotion of Research at the Technion.
Rights and permissions
About this article
Cite this article
Onn, S., Rothblum, U.G. & Tangir, Y. Edge-Directions of Standard Polyhedra with Applications to Network Flows. J Glob Optim 33, 109–122 (2005). https://doi.org/10.1007/s10898-004-4313-z
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-004-4313-z