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 ϵ>0, the
algorithm outputs in O(mlog4n/ϵ2) 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(m2log2(m)/ϵ2) running time achieved previously
by Garg and Khandekar. The LP solution can be used to obtain a fast randomized
(23+ϵ)-approximation for metric TSP which improves
upon the running time of previous implementations of Christofides' algorithm