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