The Traveling Salesman Problem Under Squared Euclidean Distances [article]

Mark de Berg and Fred van Nijnatten and René Sitters and Gerhard J. Woeginger and Alexander Wolff
2010 arXiv   pre-print
Let P be a set of points in R^d, and let α> 1 be a real number. We define the distance between two points p,q∈ P as |pq|^α, 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^α-1+√(6)^α/3 for d=2 and all α>2. We also study the variant Rev-TSP of the problem where the traveling
more » ... man is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP(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.
arXiv:1001.0236v3 fatcat:dk2ic7bm2rhalgzsfv5bwfnzwe