Four closely related minimax location problems are considered. Each involves locating a point in the plane to minimize the maximum distance (plus a possible constant) to a given finite set of points. The distance measures considered are the Euclidean and the rectilinear. In each case efficient, finite solution procedures are given. The arguments are geometrical.
Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program maxfx T Axjx T e = 1; x 0g. It is well known that the global optimum value of this QP is (1?1=!(G)), where !(G) is the clique number of G. Here, we characterize the following: 1) rst order optimality 2) second order optimality 3) local optimality 4) strict local. These characterizations reveal interesting underlying discrete structures, and are polynomial time veri able. A parametrization of the Motzkin-Strauss QP is then introduced and its properties are investigated. Finally, an extension of the Motzkin-Strauss formulation is provided for the weighted clique number of a graph and this is used to derive a maximin characterization of perfect graphs.
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