Abstract
Algorithms are given to list the vertices of polyhedra associated with network linear programs and their duals. Each algorithm has running time which is quadratic in the number of vertices of the polyhedron, and does not require that the polyhedron be either bounded or simple. The algorithms use characterizations of adjacent vertices in network and dual network LP's to perform an efficient traversal of the edge graph of the polyhedron. This contrasts with algorithms for enumerating the vertices of a general polyhedron, all of which have worst-case complexity which is exponential in the number of vertices of the polyhedron.
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References
A.V. Aho, J.E. Hopcroft and J.D. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, New York, 1974).
W. Altherr, “An algorithm for enumerating all vertices of a convex polyhedron,”Computing 15 (1975) 181–193.
D. Avis and K. Fukuda, “A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra,” Tech. Rept. SOCS-91.6, McGill University (Montréal, Qué., 1991).
M.L. Balinski, “An algorithm for finding all vertices of convex polyhedral sets,”SIAM Journal on Applied Mathematics 9 (1961) 72–88.
B. Chazelle, “An optimal convex hull algorithm and new results on cuttings (extended abstract),”Proceedings of 32nd IEEE FCOS (1991) 29–38.
M.S. Bazaraa and J.J. Jarvis,Linear Programming and Network Flows (Wiley, New York, 1977).
M.E. Dyer, “The complexity of vertex enumeration methods,”Mathematics of Operations Research 8 (1983) 381–402.
M.E. Dyer and L.G. Proll, “An algorithm for determining all extreme points of a convex polytope,”Mathematical Programming 12 (1977) 81–96.
A.J. Goldman, “Resolution and separation theorems for polyhedral convex sets,” in: H.W. Kuhn and A.W. Tucker, eds.,Linear Inequalities and Related Systems (Princeton University Press, Princeton, NJ, 1956) pp. 53–97.
E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, San Francisco, CA, 1976).
M. Manas and J. Nedoma, “Finding all vertices of a convex polyhedron,”Numerische Mathematik 12 (1968) 226–229.
T.H. Matthiess, “An algorithm for determining irrelevant constraints and all vertices in systems of linear inequalities,”Operations Research 21 (1973) 247–260.
T.H. Matthiess and D.S. Rubin, “A survey and comparison of methods for finding all vertices of convex polyhedral sets,”Mathematics of Operations Research (1980) 167–185.
T.S. Motzkin, H. Raiffa, G.L. Thompson and R.M. Thrall, “The double description method,” in: H.W. Kuhn and A.W. Tucker, eds.,Contributions to the Theory of Games, Vol. 2 (Princeton University Press, Princeton, NJ, 1953) pp. 51–73.
K.G. Murty,Linear Programming (Wiley, New York, 1983).
J.S. Provan, “Efficient recognition of matroid and 2-monotonic systems,” in: R.D. Ringeisen and F.S. Roberts, eds.,Applications of Discrete Mathematics (SIAM, Philadelphia, PA, 1988) pp. 122–134.
J.S. Provan and D.R. Shier, “A paradigm for listing (s, t)-cuts in graphs,” Tech. Rept. UNC/OR/TR91-3, University of North Carolina (Chapel Hill, NC, 1991).
R.C. Read and R.E. Tarjan, “Bounds on backtrack algorithms for listing cycles, paths and spanning trees,”Networks 5 (1975) 237–252.
D.S. Rubin, “Vertex generation and cardinality constrained linear programs,”Operations Research 23 (1975) 555–564.
D.S. Rubin, “Vertex generation methods for problems with logical constraints,”Annals of Discrete Mathematics 1 (1977) 457–466.
R. Seidel, “Constructing higher-dimensional convex hulls at logarithmic cost per face,”Proceedings of the 18th ACM STOC (1986) 404–413.
G. Swart, “Finding the convex hull facet by facet,”Journal of Algebra 6 (1985) 17–48.
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Provan, J.S. Efficient enumeration of the vertices of polyhedra associated with network LP's. Mathematical Programming 63, 47–64 (1994). https://doi.org/10.1007/BF01582058
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DOI: https://doi.org/10.1007/BF01582058