Solution for a bipartite Euclidean traveling-salesman problem in one dimension

TL;DR

This paper characterizes the optimal cycle for a bipartite Euclidean traveling-salesman problem in one dimension with convex costs, computes average costs, and explores properties like non-self-averaging and bounds relative to optimal assignments.

Contribution

It provides a complete characterization of the optimal cycle for convex cost functions in a bipartite 1D Euclidean TSP and analyzes its statistical properties.

Findings

Average optimal cost computed for squared Euclidean distances.

Optimal cycle cost is not smaller than twice the optimal assignment cost.

The average optimal cost is not a self-averaging quantity, with explicit variance calculation.

Abstract

The traveling salesman problem is one of the most studied combinatorial optimization problems, because of the simplicity in its statement and the difficulty in its solution. We characterize the optimal cycle for every convex and increasing cost function when the points are thrown independently and with an identical probability distribution in a compact interval. We compute the average optimal cost for every number of points when the distance function is the square of the Euclidean distance. We also show that the average optimal cost is not a self-averaging quantity by explicitly computing the variance of its distribution in the thermodynamic limit. Moreover, we prove that the cost of the optimal cycle is not smaller than twice the cost of the optimal assignment of the same set of points. Interestingly, this bound is saturated in the thermodynamic limit.