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We present easily verifiable conditions, under which a graph G contains nonempty vertex-disjoint induced subgraphs G1, G2 such that G is perfect if and only if G1 and G2 are. This decomposition is defined in terms of the induced subgraphs of G that are isomorphic to the chordless path with four vertices. 相似文献
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V. Chvátal 《European Journal of Combinatorics》1985,6(3):217-226
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Sylvester conjectured in 1893 and Gallai proved some 40 years later
that every finite set S of points in the plane includes two points
such that the line passing through them includes either no other point
of S or all other points of S. There are several ways of extending
the notion of lines from Euclidean spaces to arbitrary metric spaces.
We present one of them and conjecture that, with lines in metric
spaces defined in this way, the Sylvester--Gallai theorem generalizes
as follows: in every finite metric space there is a line consisting
of either two points or all the points of the space. Then we present
meagre evidence in support of this rash conjecture and finally we
discuss the underlying ternary relation of metric betweenness. 相似文献
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David Applegate Robert Bixby Vašek Chvátal William Cook 《Mathematical Programming》2003,97(1-2):91-153
Dantzig, Fulkerson, and Johnson (1954) introduced the cutting-plane method as a means of attacking the traveling salesman
problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming.
In this paper we discuss an implementation of Dantzig et al.'s method that is suitable for TSP instances having 1,000,000
or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the
general cutting-plane method in large-scale applications.
Received: December 6, 2002 / Accepted: April 24, 2003
Published online: May 28, 2003
RID="*"
ID="*" Supported by ONR Grant N00014-03-1-0040 相似文献
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A general framework for cutting-plane generation was proposed by Applegate et al. in the context of the traveling salesman problem. The process considers the image of a problem space under a linear mapping, chosen so that a relaxation of the mapped problem can be solved efficiently. Optimization in the mapped space can be used to find a separating hyperplane, if one exists, and via substitution this gives a cutting plane in the original space. We extend this procedure to general mixed-integer programming problems, obtaining a range of possibilities for new sources of cutting planes. Some of these possibilities are explored computationally, both in floating-point arithmetic and in rational arithmetic. 相似文献
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V Chvátal 《Journal of Combinatorial Theory, Series B》1985,39(3):189-199
We first establish a certain property of minimal imperfect graphs and then use it to generate large classes of perfect graphs. 相似文献
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Jarmila Chvátalová 《Discrete Mathematics》1975,11(3):249-253
If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.Pn will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. Pm × Pn will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(Pm × Pn) = min(m, n). 相似文献