Let P be a set of points in Rd, 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
3α−1+6α/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 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.Comment: 12 pages, 4 figures. (v2) Minor linguistic change