For TSP with triangle inequality, Approx-TSP-Tour is an approximation algorithm with a ratio bound of 2.

If H is approximate tour, c(H) 2c(H*)

**Proof:**

Removing an edge from a tour yields a spanning tree. Thus, if H* is the optimal tour and T is a MST(G), then c(T) c(H*).

Consider a **full walk** W of a MST with cost c(W).

Example

W = a __b__ d __b__ e b a c a

c(W) = 2c(T)
c(W) 2c(H*)

W is not a tour, but by triangle inequality, we can change w x w y to w x y, without increasing cost to yield approximate tour H. Or, use the pre-order traversal of the MST.

c(H) c(W) 2c(H*)