# Nearly linear time approximation schemes for Euclidean TSP and other geometric problems

@article{Arora1997NearlyLT, title={Nearly linear time approximation schemes for Euclidean TSP and other geometric problems}, author={Sanjeev Arora}, journal={Proceedings 38th Annual Symposium on Foundations of Computer Science}, year={1997}, pages={554-563} }

We present a randomized polynomial time approximation scheme for Euclidean TSP in R/sup 2/ that is substantially more efficient than our earlier scheme (1996) (and the scheme of Mitchell (1996)). For any fixed c>1 and any set of n nodes in the plane, the new scheme finds a (1+1/c)-approximation to the optimum traveling salesman tour in O(n(logn)/sup O(c)/) time. (Our earlier scheme ran in n/sup O(C)/ time.) For points in R/sup d/ the algorithm runs in O(n(logn)/sup (O(/spl radic/dc)/d-1)) time…

## 181 Citations

### A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem

- Mathematics, Computer ScienceSIAM J. Comput.
- 2007

This paper provides a randomized approximation scheme for the k-median problem when the input points lie in the d-dimensional Euclidean space and develops a structure theorem to describe hierarchical decomposition of solutions.

### A Fast Approximation Algorithm for TSP with Neighborhoods

- Computer ScienceNord. J. Comput.
- 1999

This paper presents a simple and fast algorithm that, given a start point, computes a TSPN tour of length O(log k) times the optimum in time O(n+k log k).

### Approximation Algorithms for the Euclidean Traveling Salesman Problem with Discrete and Continuous Neighborhoods

- Computer Science, MathematicsInt. J. Comput. Geom. Appl.
- 2009

This work seeks to find a tour of minimum length which visits at least one point in each region of Euclidean traveling salesman problem with discrete neighborhoods, and gives an O(α)-approximation algorithm for the case when the regions are disjoint and α-fat, with possibly varying size.

### Improved Approximation Schemes for Geometrical Graphs Via Spanners and Banyans

- Computer ScienceSTOC 1998
- 1998

The algorithms are based on using low-weight Eu-clidean spanner graphs in conjunction with the hierarchical structure theorems that serve as the basis of Arora's work and it is shown that spanners can in principle be made orders of magnitude shorter and simpler by allowing Steiner points.

### A Polynomial Time Approximation Scheme for Euclidean Minimum Cost k-Connectivity

- Mathematics, Computer ScienceICALP
- 1998

It is observed that the time cost of the derandomization of the PTA schemes for Euclidean optimization problems in ℝd derived by Arora can be decreased by a multiplicative factor of Ω(n d−1 ).

### Approximating Asymmetric TSP in exponential Time

- Computer Science, MathematicsInt. J. Found. Comput. Sci.
- 2014

A very simple algorithm is proposed that, for any 0 < e < 1, finds (1+e)-approximation to asymmetric TSP in 2ne−1 time and e−1 · poly(n, log M) space.

### On the Efficiency of Polynomial Time Approximation Schemes

- Computer Science, MathematicsInf. Process. Lett.
- 1997

### A divide-and-conquer algorithm for min-cost perfect matching in the plane

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

A variant of Edmonds algorithm that uses geometric divide-and-conquer, so that in the conquer step the authors need only O(/spl radic/n) phases, and it is shown that a single phase can be implemented in O(n log/sup 5/ n) time.

### Simplicity and hardness of the maximum traveling salesman problem under geometric distances

- Computer ScienceSODA '99
- 1999

A simple algorithm with O(n) running time for computing the length of a longest tour for a set of points in the plane with rectilinear distances and gets NP-hardness of the Maximum Scatter TSP for geometric instances, where the objective is to find a tour that maximizes the shortest edge.

### Approximation schemes for Euclidean k-medians and related problems

- Mathematics, Computer ScienceSTOC '98
- 1998

An approximation scheme for the plane that for any c > 0 produces a solution of cost at most 1+ 1/c times the optimum and runs in time O(n) and generalizes to some problems related to k-median.

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