If P NP, there is no poly-time approximation algorithm with ratio bound p 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.