On The Approximability Of The Traveling Salesman Problem

We show that the traveling salesman problem with triangle inequality cannot be approximated with a ratio better than $\frac{{117}} {{116}}$ when the edge lengths are allowed to be asymmetric and $\frac{{220}} {{219}}$ when the edge lengths are symmetric, unless P=NP. The best previous lower bounds were $\frac{{2805}} {{2804}}$ and $\frac{{3813}} {{3812}}$ respectively. The reduction is from Håstad’s maximum satisfiability of linear equations modulo 2, and is nonconstructive.