The Traveling Salesman Problem under Squared Euclidean Distances

Let P be a set of points in R, and let α > 1 be a real number. We define the distance between two points p, q ∈ P as |pq|, where |pq| denotes the standard Euclidean distance between p and q. We denote the traveling salesman problem under this distance function by Tsp(d,α). We design a 5-approximation algorithm for Tsp(2,2) and generalize this result to obtain an approximation factor of 3 + √ 6 α /3 for d = 2 and all α > 2. We also study the variant Rev-Tsp of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for RevTsp(2, α) with α > 2, and we show that Rev-Tsp(d,α) is apx-hard if d > 3 and α > 1. The apx-hardness proof carries over to Tsp(d, α) for the same parameter ranges.