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We show that the average cost for the traveling-salesman problem in twodimensions, which is the archetypal problem in combinatorial optimization, inthe bipartite case, is simply related to the average cost of the assignmentproblem with the same Euclidean, increasing, convex weights. In this way weextend a result already known in one dimension where exact solutions areavalaible. The recently determined average cost for the assignment when thecost function is the square of the distance between the points providestherefore an exact prediction $$\overline{E_N} = \frac{1}{\pi}\, \log N$$ forlarge number of points $2N$. As a byproduct of our analysis also the loopcovering problem has the same optimal average cost. We also explain why thisresult cannot be extended at higher dimensions. We numerically check the exactpredictions.

Exact value for the average optimal cost of bipartite traveling-salesman and 2-factor problems in two dimensions