An O(log n/log log n)-approximation Algorithm for the Asymmetric Traveling Salesman Problem

Abstract

We consider the Asymmetric Traveling Salesman problem for costs satisfying the triangle inequality. We derive a randomized algorithm which delivers a solution within a factor O(log n/ log log n) of the optimum with high probability.

The Asymmetric Traveling Salesman Problem on Graphs with Bounded Genus

“…Our result builds on a central lemma in Asadpour et al [3] that shows for finding a constantfactor approximation algorithm for ATSP, it is sufficient to find a "thin" tree in the fractional solution. Roughly speaking, a tree is -thin with respect to a graph G, if it does not contain more than an -fraction of the edges of G across any cut.…”

“…On the other hand, our approach for finding a tree and establishing its thinness is quite different from Asadpour et al [3]: the embedding of the graph and its geometric dual will be crucial in finding a tree and proving its thinness. In particular, we take advantage of the correspondence between the cutsets of the graph G and cycles of the dual graph G * .…”

A Randomized Rounding Approach to the Traveling Salesman Problem

“…We propose the same algorithm as in Asadpour et al [4] for TSP. Let x be the optimum solution of the Held-Karp linear programming relaxation.…”

“…In the analysis of this algorithm for asymmetric TSP [4], Asadpour et al use the negative correlation between the edges of random spanning trees to obtain concentration results on the distribution of edges across a cut. For this work, we have to use even stronger virtues of negative dependence [34].…”

“…Here, our idea is similar to Asadpour et al [4]. The algorithm selects a spanning tree randomly from G, the support graph of the solution of LP subtour .…”