The Algebraic Degree of Geometric Optimization Problems

AJ1STRACT In this paper we apply Galois theoretic algebraic methods to certain fundamental geometric optimization problems whose recognition versions are not even known to be in the class NP. In particular we show that the classic Weber problem, the Line-restricted' Weber probe lem and the 3-Dimension version of this problem are in general not solvable by radicals over the field of rationals. One: direct consequence of these results is that for these geometric optimization problems there exists no exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic opf;rations and the extraction of e h roots. This leaves only numeric or symbolic approximations to the SOlutions, where the complexity of the approximations is shown to be primarily a function of the algebraic degree of the optimum solution point.