  Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a
real number. We define the distance between two points $p,q\in P$ as
$|pq|^{\alpha}$, where $|pq|$ 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
$3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$.
  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 $\alpha\ge2$, and we show that Rev-TSP$(d,
\alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries
over to TSP$(d, \alpha)$ for the same parameter ranges.
