Theorem 37.2

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) $\leq$ 2c(H*)

Proof:

If H* is the optimal tour and T is a MST(G), then c(T) $\leq$ 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) $\longrightarrow$ c(W) $\leq$ 2c(H*)

W is not a tour, but by triangle inequality, we can change w $\rightarrow$ x $\rightarrow$ w $\rightarrow$ y to w $\rightarrow$ x $\rightarrow$ y, without increasing cost to yield approximate tour H.

c(H) $\leq$ c(W) $\leq$ 2c(H*)