# Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

## TL;DR

This paper presents a nearly linear time randomized approximation scheme for the Held-Karp bound in metric TSP, significantly improving computational efficiency and enabling faster approximate solutions.

## Contribution

It introduces a nearly linear time algorithm for approximating the Held-Karp bound, surpassing previous quadratic time methods, and facilitates faster TSP approximations.

## Key findings

- Achieves $O(m \log^4 n / \epsilon^2)$ runtime for approximation
- Provides a $(1+\epsilon)$-approximate solution to the Held-Karp bound
- Enables a faster $(rac{3}{2} + \epsilon)$-approximation for metric TSP

## Abstract

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $\epsilon > 0$, the algorithm outputs in $O(m \log^4n /\epsilon^2)$ time, with high probability, a $(1+\epsilon)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/\epsilon^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + \epsilon\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm.