Are there any known exact algorithms for Euclidean TSP that take advantage of the inherent structure of the problem?  Do any of these algorithms have better asymptotics than $O(2^n n^2)$ of a DP solution via Held-Karp?  I am also interested in simplicity of implementation (I've seen a 10-line implementation of Held-Karp in Python).Is this still the current state of the art?
Euclidean TSP algorithms
tsp;dynamic programming