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

We give a nearly linear-time randomized approximation scheme for the Held-Karp bound [22] for Metric-TSP. Formally, given an undirected edge-weighted graph G = (V, ε) on m edges and ε > 0, the algorithm outputs in O(m log
⁴
n/ε
²
) time, with high probability, a (1 + ε)-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)/ε
²
) running time achieved previously by Garg and Khandekar.