# An Improved Lower Bound for the Traveling Salesman Constant

## TL;DR

This paper improves the known lower bound for the Traveling Salesman constant, which describes the asymptotic length of the shortest tour through random points in a unit square, from 0.625 to 0.6277.

## Contribution

The authors enhance the lower bound estimate of the Traveling Salesman constant using an approach based on Steinerberger's method, advancing previous results.

## Key findings

- Lower bound for the TSP constant is improved to 0.6277
- The approach builds on Steinerberger's method from 2015
- The result refines the understanding of TSP tour length asymptotics

## Abstract

Let $X_1, X_2, \dots, X_n$ be independent uniform random variables on $[0,1]^2$. Let $L(X_1, \dots, X_n)$ be the length of the shortest Traveling Salesman tour through these points. It is known that there exists a constant $\beta$ such that $$\lim_{n \to \infty} \frac{L(X_1, \dots, X_n)}{\sqrt{n}} = \beta$$ almost surely (Beardwood 1959). The original analysis in (Beardwood 1959) showed that $\beta \geq 0.625$. Building upon an approach proposed in (Steinerberger 2015), we improve the lower bound to $\beta \geq 0.6277$.