Exact value for the average optimal cost of the bipartite traveling salesman and two-factor problems in two dimensions

We show that the average optimal cost for the traveling salesman problem in two dimensions, which is the archetypal problem in combinatorial optimization, in the bipartite case, is simply related to the average optimal cost of the assignment problem with the same Euclidean, increasing, convex weights. In this way we extend a result already known in one dimension where exact solutions are available. The recently determined average optimal cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction EN=1πlogN for large number of points 2N. As a by-product of our analysis, also the loop covering problem has the same optimal average cost. We also explain why this result cannot be extended to higher dimensions. We numerically check the exact predictions.