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PublisherThe University of Arizona.
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AbstractCenter location on cactus graphs. The p-center problem has been shown to be NP-hard for case of a general graph, yet polynomial algorithms exist for the case of a tree graph. Specifically, we consider "cactus graphs" where each edge is contained in at most one cycle. We show that the p-center problem on this class can be solved in polynomial time using a decomposition algorithm. We partition the graph into a set of subgraphs which are then solved sequentially. The solutions to the subgraphs are linked by a single parameter. The algorithm runs in polynomial time. Locating capacity limited centers on trees. The uncapacitated p-center problem on trees is solvable in polynomial time. We extend this result to include the case where each center can serve a limited number of customers and show that the capacitated p-center on trees can be solved in polynomial time when the capacities are identical. The algorithm consists of solving a capacitated covering problem and then using search routines to find the optimal domination radius. Center location on spheres. We discuss the unweighted center location problem. The following results are presented: (i) An O(n) time algorithm to solve the 1-center problem if the vertices are on one half of the sphere, and an O(n) time algorithm to check whether this condition holds. Both algorithms are based on presenting the problems as 3-dimensional convex programming problems with linear constraints and applying a pruning technique to find the optimum in O(n) time. (ii) An O(n$\sp3$ log n) time algorithm for the 2-center problem on the whole sphere. (iii) A reduction to show that the general p-center problem on a sphere is NP-hard. Locating hyperplanes on hypercubes. In linear regression models we are interested in locating a (d-1) dimensional hyperplane that will be as "close" as possible to existing vertices in the d-dimensional hypercube. The least squares criterion is usually applied for the linear fitting problem; while fitting according to the least absolute value ("minisum") seems to be "complicated". We solve fitting problems with the minisum criterion and present linear time algorithms when the dimension d is fixed. (Abstract shortened with permission of author.)
Degree ProgramSystems and Industrial Engineering