The traveling salesman problem is one of the most fundamental optimization problems. Given n cities and pairwise distances, it is the problem of finding a tour of minimum distance that visits each city once. In spite of significant research efforts, current techniques seem insufficient for settling the approximability of the traveling salesman problem.  This status is particularly intriguing as a natural and several-decade-old linear programming relaxation is believed to give better guarantees than we are currently able to prove!

In this mini-course, we will overview of old and new approaches for settling this question. We shall, in particular, talk about recent developments for the asymmetric traveling salesman problem.