Finding anEuclideananti-k-centrum locationofasetofpoints

An obnoxiousfacilityistobelocatedinsideapolygonalregionoftheplane,maximizingthesumofthe k smallest weightedEuclideandistancesto n given points,eachprotectedbysomepolygonalforbidden region. Fortheunweightedcaseand k fixed an O(n2 log n) timealgorithmispresented.Fortheweighted case athoroughstudyoftherelevantstructureofthemultiplicativelyweightedorder-k-Voronoi diagram leads tothedesignofan O(kn3 + n3 log n) timealgorithmforfindinganoptimalsolutiontotheanti- t-centrum problemforevery t = 1,..., k, simultaneously.