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

TL;DR

This paper derives an exact formula for the average optimal cost of bipartite traveling-salesman and 2-factor problems in two dimensions, linking it to the assignment problem and confirming predictions through numerical checks.

Contribution

It establishes a precise relationship between the bipartite TSP and the assignment problem in 2D, providing exact average cost formulas and explaining dimensional limitations.

Findings

Average cost for bipartite TSP in 2D is proportional to (1/π) log N.

The same cost applies to the loop covering problem.

Numerical results confirm the theoretical predictions.

Abstract

We show that the average 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 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 avalaible. The recently determined average cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction $$\overline{E_N} = \frac{1}{\pi}\, \log N$$ for large number of points $2N$. As a byproduct of our analysis also the loop covering problem has the same optimal average cost. We also explain why this result cannot be extended at higher dimensions. We numerically check the exact predictions.