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Ramsey Numbers of K_m versus (n,k)-graphs and the Local Density of Graphs not Containing a K_m
(1999)

In this paper generalized Ramsey numbers of complete graphs K_m versus the set langle ,n,k angle of (n,k)-graphs are investigated. The value of r(K_m,langle n,k angle) is given in general for (relative to n) values of k small compared to n using a correlation with Turan numbers. These generalized Ramsey numbers con be used to determine the local densities of graphs not containing a subgraph K_m.

The Weber problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new facility X* in R^2 such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by polyhedral gauges. A variation of this problem is obtained if barriers are introduced which are convex polygonal subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers like lakes, military regions, national parks or mountains are frequently encountered in practice.From a mathematical point of view barrier problems are difficult, since the prensence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a descretization result: One of the grid points in the grid defined by the existing facilities and the fuundamental directions of the gauge distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm.

In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facility where a set of polygonal barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the Median and Center objective functions). An exact Algorithm and a heuristic solution procedure based on this reduction result are developed.

The Weber Problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new fcility X* such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. A variation of this problem is obtained of the existing facilities are situated on two sides of a linear barrier. Such barriers like rivers, highways, borders or mountain ranges are frequently encountered in practice. Structural results as well as algorithms for this non-convex optimization problem depending on the distance function and on the number and location of passages through the barrier are presented. A reduction to convex optimization problems is used to derive efficient algorithms.