## Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems (1997)

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Citations: | 95 - 3 self |

### BibTeX

@MISC{Arora97nearlylinear,

author = {Sanjeev Arora},

title = {Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems},

year = {1997}

}

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### Abstract

We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). 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(log n) O(c) ) time. (Our earlier scheme ran in n O(c) time.) For points in ! d the algorithm runs in O(n(log n) (O( p dc)) d\Gamma1 ) time. This time is polynomial (actually nearly linear) for every fixed c; d. Designing such a polynomial-time algorithm was an open problem (our earlier algorithm in [2] ran in superpolynomial time for d 3). The algorithm generalizes to the same set of Euclidean problems handled by the previous algorithm, including Steiner Tree, k-TSP, k-MST, etc, although for k-TSP and k-MST the running time gets multiplied by k. We also use our ideas to design nearly-linear time approximation schemes for Euclidean vers...

### Citations

1513 | Reducibility among combinatorial problems - Karp - 1972 |

579 | Optimization, Approximation, and Complexity Classes - Papadiitriou, Yannakakis - 1991 |

353 | An effective heuristic algorithm for the traveling salesman problem - Lin, Kernighan |

332 | The traveling salesman problem - Lawler, Lenstra, et al. - 1995 |

288 |
The traveling salesman problem: A case study
- Johnson, McGeoch
- 1997
(Show Context)
Citation Context ...ward NSF CCR-9502747 and an Alfred Sloan Fellowship. Email: arora@cs.princeton.edu in nearly linear time and often produce tours whose cost is within a few percent of the optimum (Johnson and McGeoch =-=[13]-=-). Potentially, this paradox could have a simple explanation since many "real-life" instances are geometric (see for example the TSPLIB library [24], a testbed for much experimental work on the TSP), ... |

287 |
Worst-case analysis of a new heuristics for the traveling salesman problem
- Christofides
- 1976
(Show Context)
Citation Context ...entioned paradox by designing an excellent approximation algorithm for geometric TSP. However, until recently, we knew of no polynomial-time algorithm better than the Christofides algorithm from 1976 =-=[7]-=-, which provably computes a tour of cost at most 1:5 times the optimum. Recently, Euclidean TSP was shown to have a polynomial time approximation scheme or PTAS (Arora [2] and some time later, Mitchel... |

287 | TSPLIB – a Traveling Salesman Problem Library - Reinelt - 1991 |

205 | Proof Verification and Intractability of Approximation Problems - Arora, Lund, et al. - 1992 |

190 |
Fast approximation algorithms for the knapsack and sum of subset problems
- Ibarra, Kim
- 1975
(Show Context)
Citation Context ...s algorithm seems to work only in !2.) With the discovery of this algorithm, Euclidean TSP joined a select group of NP-hard problems that have PTAS's, a group that includes Subset-Sum (Ibarra and Kim =-=[12]-=-) and Bin-Packing (de la Vega and Lueker [8]; see also Karmarkar and Karp [14]) Though theoretically significant, the recent PTAS for Euclidean TSP is impractical. The constant inside the big-O is abo... |

165 | Guillotine subdivisions approximate polygonal subdivisions: Part II---a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems - Mitchell - 1999 |

151 | Data structures for weighted matching and nearest common ancestors with linking - Gabow - 1990 |

121 | An efficient approximation scheme for the one–dimensional bin–packing problem - Karmarkar, Karp - 1982 |

101 | Probabilistic analysis of partitioning algorithms for the traveling-salesman problem - Karp - 1977 |

93 |
Polynomial-time approximation schemes for Euclidean TSP and other geometric problems
- Arora
- 1998
(Show Context)
Citation Context ...anjeev Arora \LambdasPrinceton University Abstract We present a randomized polynomial time approximation scheme for Euclidean TSP in !2 that is substantially more efficient than our earlier scheme in =-=[2]-=- (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1c )-approximation to the optimum traveling salesman tour in O(n(log n)O(c)) tim... |

85 | Some np-complete geometric problems
- Garey, Graham, et al.
- 1976
(Show Context)
Citation Context ...reat those squares as independent instances is justified! 4 Conclusions Can the running time of our algorithm be improved? Since Euclidean TSP is strongly NP-hard (this follows from the reductions in =-=[11, 22]-=-, for example), any algorithm that computes (1 + c)- approximations must have a running time with some exponential dependence on c (unless of course, NPcomplete problems can be solved in subexponentia... |

81 | Approximation algorithms for geometric problems, Approximation Algorithms for NP-hard Problems
- Bern, Eppstein
- 1997
(Show Context)
Citation Context ...nadjacent edges that cover all vertices. A PTAS was described for the first three problems in [2]; prior to that they had constant factor approximation algorithms (see the survey by Bern and Eppstein =-=[5]-=-). Theorem 1 (Main) There is a randomized PTAS for all the above problems that computes a (1 + 1=c)- approximation on planar instances in O(n(log n)O(c)) time for TSP, Steiner Tree and EMCPM, and in O... |

63 | Geometry Helps in Matching - Vaidya - 1989 |

60 | S.-H.Teng. Parallel construction of quadtree and quality triangulations
- Bern, Eppstein
- 1993
(Show Context)
Citation Context ...ructure than the original quadtree. However, since both have depth O(log n), they can be constructed very efficiently using a sorting-based algorithm in O(n log2 n) time (even faster algorithms exist =-=[6]-=-). Our algorithm picks a random shift and constructs the corresponding shifted quadtree. Our Structure Theorem below guarantees that with high probability (over the choice of the random shifts), there... |

57 | Iterated nearest neighbors and finding minimal polytopes
- Eppstein, Erickson
- 1994
(Show Context)
Citation Context ...rst it computes in nearly-linear time a crude approximation to OP T that is correct within a factor n. (Algorithms to compute such approximations were known before our work; see [27] for matching and =-=[9]-=- for k-MST and k-TSP. The latter algorithm runs in O(nk log n) time.) Let A ^ OP T \Deltasn be this approximation. The procedure lays a grid of granularity A=8cn2 in the plane and moves every node to ... |

48 | An approximation scheme for planar graph TSP - Grigni, Koutsoupias, et al. - 1995 |

43 | When Hamming meets Euclid: the approximability of geometric TSP and MST - Trevisan - 1997 |

22 | Euclidean TSP is NP-complete - Papadimitriou - 1977 |

3 |
Euclidean MST and bichromatic closest pairs. Discrete and
- Agarwal, Edelsbrunner, et al.
- 1991
(Show Context)
Citation Context ...e been studied before, with an aim to exploit geometric structure and derive algorithms that are faster than the corresponding algorithms for general graphs. For any fixed dimension d, Agarwal et al. =-=[1]-=- have designed a ~O(n4=3)-algorithm for Euclidean MST and Vaidya [26] has designed a~ O(n2:5)-algorithm for matching. Vaidya [27] has also designed approximation schemes that compute (1 + 1=c)-approxi... |

2 | Approximate minimum weight matching on k-dimensional spaces - Vaidya - 1989 |

2 | Is the m parameter in Arora's TSP algorithm the best possible? Junior project - Wang - 1996 |