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,E) 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.