For traveling salesmen problem there is Christofides algorithm which gives result that is no longer than 3/2 of optimal one. Theoretical bound on inapproximabil

For traveling salesmen problem there is Christofides algorithm which gives result that is no longer than 3/2 of optimal one. Theoretical bound on inapproximability of symmetrical TSP currently is 185/184. https://arxiv.org/abs/1303.6437 https://arxiv.org/abs/1303.6437