THE TRAVELING SALESMAN PROBLEM UNDER SQUARED EUCLIDEAN DISTANCES

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