# An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem

@article{Asadpour2010AnON, title={An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem}, author={Arash Asadpour and Michel X. Goemans and Aleksander Madry and Shayan Oveis Gharan and Amin Saberi}, journal={Oper. Res.}, year={2010}, volume={65}, pages={1043-1061} }

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.

## 90 Citations

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The traveling salesman problem (TSP) is a well-known, commonly studied, and very important NP-hard optimization problem in theoretical computer science, of which symmetric TSP is a special case, and this work will be considering a slightly more general TSP variant, asymmetric T SP, which allows c({ (u, v) 6= c({(v, u)}).

## References

SHOWING 1-10 OF 57 REFERENCES

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

- Computer ScienceSODA
- 2010

A randomized algorithm is derived which delivers a solution within a factor O(log n/ log log n) of the optimum of the Asymmetric Traveling Salesman problem with high probability.

A Randomized Rounding Algorithm for the Asymmetric Traveling Salesman Problem

- Computer ScienceArXiv
- 2009

An algorithm for the asymmetric traveling salesman problem on instances which satisfy the triangle inequality is presented and achieves approximation ratio O(log n).

On the worst-case performance of some algorithms for the asymmetric traveling salesman problem

- Computer ScienceNetworks
- 1982

This work considers the asymmetric traveling salesman problem for which the triangular inequality is satisfied, and constructs examples to show that the worst-case ratio of length of tour found to minimum length tour is (n) for n city problems.

The asymmetric traveling salesman problem on graphs with bounded genus

- MathematicsSODA '11
- 2011

We give a constant factor approximation algorithm for the asymmetric traveling salesman problem when the support graph of the solution of the Held-Karp linear programming relaxation has bounded…

A new approximation algorithm for the asymmetric TSP with triangle inequality

- Computer ScienceSODA '03
- 2003

A polynomial time factor 0.999·log n approximation algorithm is presented for the asymmetric traveling salesperson problem with triangle inequality and it is shown that the solution to this problem is simple and efficient.

On the Integrality Ratio for the Asymmetric Traveling Salesman Problem

- MathematicsMath. Oper. Res.
- 2006

We improve the lower bound on the integrality ratio of the Held-Karp bound for asymmetric TSP with triangle inequality from 4/3 to 2.

A Randomized Rounding Approach to the Traveling Salesman Problem

- Computer Science, Mathematics2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
- 2011

This work gives a (3/2-\eps_0)-approximation algorithm that finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree.

Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs

- Computer Science, MathematicsAPPROX-RANDOM
- 2007

In metric asymmetric traveling salesperson problems the input is a complete directed graph in which edge weights satisfy the triangle inequality, and one is required to find a minimum weight walk…

Randomized rounding: A technique for provably good algorithms and algorithmic proofs

- Computer Science, MathematicsComb.
- 1987

A randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0–1 problem is given and can be extended to provide bounds on the disparity between the rational and 0-1 optima for a given problem instance.

A convex relaxation for the asymmetric TSP

- Computer Science, MathematicsSODA '99
- 1999

The main result is that one can find a fractional solution to the relaxation that is very sparse ( with < 3n edges) and it is shown that in the special case when the underlying graph is Hamiltonian with edge lengths 1 and the (inand out-) degrees of every vertex are each at most 2, a solution toThe relaxation can be rounded to an integral solution (a tour) whose length is at most twice the optimum.