Inapproximability of the Euclidean power-p Steiner tree problem

For a real number p ≥ 2, an integer k > 0 and a set of terminals X in the plane, the Euclidean power-p Steiner tree problem asks for a tree interconnecting X and at most k Steiner points such that the sum of the p-th powers of the edge lengths is minimised. We show that this problem is in the complexity subclass exp-APX (but not poly-APX) of NPO. We then demonstrate that the approximation algorithm obtained by placing degree-2 Steiner points on the edges of the minimum spanning tree has a performance ratio κ, where √ 3 p−2 ( 1 + 21−p ) ≤ κ ≤ 3·2p−1.