# A Randomized Rounding Approach to the Traveling Salesman Problem

@article{Gharan2011ARR, title={A Randomized Rounding Approach to the Traveling Salesman Problem}, author={Shayan Oveis Gharan and Amin Saberi and Mohit Singh}, journal={2011 IEEE 52nd Annual Symposium on Foundations of Computer Science}, year={2011}, pages={550-559} }

For some positive constant \eps_0, we give a (3/2-\eps_0)-approximation algorithm for the following problem: given a graph G_0=(V,E_0), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G_0. The result improves on the 3/2-approximation algorithm due to Christofides [C76] for this special case. Similar to Christofides, our algorithm finds a spanning…

## 177 Citations

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## References

SHOWING 1-10 OF 52 REFERENCES

The Traveling-Salesman Problem and Minimum Spanning Trees

- Computer ScienceOper. Res.
- 1970

It is shown that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers.

LP-Based Approximation Algorithms for Traveling Salesman Path Problems

- Computer ScienceArXiv
- 2011

It is shown that the recent result of Oveis Gharan, Saberi and Singh on the traveling salesman circuit problem under the unit-weight graphical metric can be modified for the path case to complement Hoogeveen's algorithm in the critical case, providing an improved performance guarantee of (5/3 - epsilon).

Minimum Bounded Degree Spanning Trees

- Mathematics, Computer Science2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
- 2006

The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree.

Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem

- Business, Computer ScienceOperations Research Forum
- 2022

An O(n3) heuristic algorithm is described for solving d-city travelling salesman problems (TSP) whose cost matrix satisfies the triangularity condition and a worst-case analysis of this heuristic shows that the ratio of the answer obtained to the optimum TSP solution is strictly less than 3/2.

TSP on Cubic and Subcubic Graphs

- Mathematics, Computer ScienceIPCO
- 2011

It is proved that, as an upper bound, the 4/3 conjecture is true for this problem on cubic graphs, and a polynomial-time 7/5-approximation algorithm and a 7/ 5 bound on the integrality gap are obtained.

Approximating Graphic TSP by Matchings

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

A framework for approximating the metric TSP based on a novel use of matchings that allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.

Analyzing the Held-Karp TSP Bound: A Monotonicity Property with Application

- Computer ScienceInf. Process. Lett.
- 1990

Minimum-weight two-connected spanning networks

- MathematicsMath. Program.
- 1990

We consider the problem of constructing a minimum-weight, two-connected network spanning all the points in a setV. We assume a symmetric, nonnegative distance functiond(·) defined onV × V which…

A linear-time approximation scheme for planar weighted TSP

- Computer Science46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
- 2005

An algorithm requiring O(c/sup 1/c2/ n) time to find an /spl epsi/-optimal traveling salesman tour in the metric defined by a planar graph with nonnegative edge-lengths is given.

An Improved Upper Bound for TSP in Cubic 3-Connected Graphs

- Mathematics, Computer Science
- 2005

One main result is an approximation algorithm for the minimum TSP on this class of graphs with an approximation factor better than the general 3/2.