Theorem 37.3

If P $\neq$ NP, there is no poly-time approximation algorithm with ratio bound p $\geq$ 1 for the general TSP (i.e., no triangle inequality).

Proof:

If such an algorithm A exists, then we can use A to solve the Hamiltonian Cycle problem, which is NP-Complete, in polynomial time.

From graph G for Hamiltonian Cycle problem construct complete graph G' = (V,E'), where edges appearing in G have cost 1, and remaining edges have cost p|V| + 1.

If HC in G, then there is a tour of cost |V| in G', and A must return it to satisfy its ratio bound of p. If no HC in G, the TSP tour costs at least

(p|V| + 1) + (|V| - 1) > p|V|.

Thus, can determine if HC in G based on whether TSP tour cost is |V|. But, unless P = NP, such an algorithm cannot exist, because it solves an NP-complete problem in polynomial time.