Certified Approximation Algorithms for the Fermat Point

Given a set of k points A ⊆ R together with a positive weight function w : A → R>0, the Fermat distance function is φ(x) = ∑ a∈A w(a)‖x− a‖. A classic problem in facility location is finding the Fermat point x∗, the point that minimizes the function φ. We present algorithms to compute an ε-approximation of the Fermat point x∗, that is, a point x̃ satisfying ‖x̃−x∗‖ < ε. Our algorithms are based on the subdivision paradigm, which we combine with Newton methods, used for speed-ups and certification. Our algorithms are certified in the sense of interval methods.