Script of Lecture 3 , Approximation Algorithms Summer term 2017

We first scale the instance. If we see two points x, y as a vector, we can replace them by αx and αy. By the properties of a norm, w(αx, αy) = ‖αx − αy‖ = α‖x − y‖. Therefore all distances are scaled by α. We choose an α > 0 such that all vertices fit exactly into an axis-parallel n2 × n2 square (where n = |V |), i. e., all vertices fit into the square and there are two vertices that lie on opposite boundaries of the square. We assume n = 2k ′ for an integer k′ (add up to |V | new vertices until a power of 2 is reached). Then L := n2 = 2k for k = 2 log2 n. Within the L × L box, we move each vertex to the closest center of an integer 1 × 1 square (break ties arbitrarily). This step is called perturbation.