## Polynomial time approximation schemes for Euclidean TSP and other geometric problems (1996)

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Venue: | In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96 |

Citations: | 336 - 3 self |

### BibTeX

@INPROCEEDINGS{Arora96polynomialtime,

author = {Sanjeev Arora},

title = {Polynomial time approximation schemes for Euclidean TSP and other geometric problems},

booktitle = {In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96},

year = {1996},

pages = {2--11}

}

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### Abstract

Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)-approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-approximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

### Citations

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295 |
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287 |
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Citation Context ...dependently from the unit square, then the fixed dissection heuristic with high probability finds tours whose cost is within a factor 1 + 1/c of optimal (where c>1 is arbitrarily large). Christofides =-=[16]-=- designed an approximation algorithm that runs in polynomial time and for every instance of metric TSP computes a tour of cost at most 3/2 times the optimum. Two decades of research failed to improve ... |

287 |
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Citation Context ...hange heuristics such as K-OPT or LinKernighan [45] seem to compute very good tours on “real-life” TSP instances [34; 10]. Most of these “real-life” instances (e.g., the ones in the well-known TSPLIB =-=[55]-=- testbed) are either Euclidean or derived from Euclidean instances. Even for such restricted classes of instances, efforts to theoretically demonstrate the goodness of local-exchange heuristics have f... |

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Citation Context ...]). So is approximating the optimum within any constant factor (Sahni and Gonzalez [56]). There are also other reasons to believe in the TSP’s nastiness (cf. DP completeness [52] and PLS-completeness =-=[35]-=-). But TSP instances arising in practice are usually quite special, so the hardness results may not necessarily apply to them. In metric TSP the nodes lie in a metric space (i.e., the distances satisf... |

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179 |
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166 |
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Citation Context ... by L 2 = O(n 2 ). 2.2 Proof of the Structure Theorem In this section we prove Theorem 2. Lemmas 3 and 4 will be important ingredients of the proof. Lemma 3 is implicit in prior work on Euclidean TSP =-=[9; 39]-=-, and can safely be called a “folk theorem.” However, we have never seen it stated precisely as it is stated here. When we later use this lemma, the “closed path” π of the hypothesis will be a salesma... |

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Citation Context ...eeks later, Mitchell [49] independently discovered an n O(c) time approximation scheme for points in ℜ 2 . His algorithm useds ideas from his earlier constant-factor approximation algorithm for k-MST =-=[48]-=-. It relies on the geometry of the plane and does not seem to generalize to higher dimensions. In January 1997 the author discovered the nearly-linear-time algorithm described in this paper. The key i... |

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Citation Context ...ertain restrictions on the number of vehicles; they use our Structure Theorem. Czumaj and Lingas [17] apply our techniques to the min-cost k-connectivity problem for small k. Arora, Raghavan, and Rao =-=[6]-=- have extended our techniques to design approximation schemes for Euclidean k-median and facility location. The Patching Lemma does not hold for these problems, and they use only (a modification of) t... |

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Citation Context ...] conjecture said it can’t be any further from the optimum; this conjecture was recently proved by Du and Hwang [19]. A spate of research activity in recent years (starting with the work of Zelikovsky=-=[64]-=-) has provided better algorithms, with an approximation ratio around 1.143 [65]. The metric case is MAX-SNP-hard [13]. k-TSP:. Given n nodes in ℜ d and an integer k>1, find the shortest tour that visi... |

101 |
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Citation Context ...n Euclidean TSP is NP-hard (Papadimitriou [50], Garey, Graham, and Johnson [24]). Therefore algorithm designers were left with no choice but to consider more modest notions of a “good” solution. Karp =-=[39]-=-, in a seminal work on probabilistic analysis of algorithms, showed that when the n nodes are picked uniformly and independently from the unit square, then the fixed dissection heuristic with high pro... |

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Citation Context ...y-optimum tour for K-OPT (for4 · S. Arora 1.0.0.1 History.. The current paper evolved out of preliminary results obtained in January 1996, culminating in a submission to IEEE FOCS 1996 in April 1996 =-=[3]-=-. The running time of the algorithm then was n O(c) in ℜ 2 and n Õ(cd−1 log d−2 n) in ℜ d . A few weeks later, Mitchell [49] independently discovered an n O(c) time approximation scheme for points in ... |

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Citation Context ...ation is NP-hard (Karp [38]). So is approximating the optimum within any constant factor (Sahni and Gonzalez [56]). There are also other reasons to believe in the TSP’s nastiness (cf. DP completeness =-=[52]-=- and PLS-completeness [35]). But TSP instances arising in practice are usually quite special, so the hardness results may not necessarily apply to them. In metric TSP the nodes lie in a metric space (... |

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Citation Context ...for some d) and distance is defined using the ℓ2 norm. Note that Euclidean TSP is a subcase of metric TSP. Unfortunately, even Euclidean TSP is NP-hard (Papadimitriou [50], Garey, Graham, and Johnson =-=[24]-=-). Therefore algorithm designers were left with no choice but to consider more modest notions of a “good” solution. Karp [39], in a seminal work on probabilistic analysis of algorithms, showed that wh... |

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Citation Context ... the case with previous (constant factor) approximation algorithms for k-TSP and k-MST. Those algorithms relied on the geometry of the plane and broke down even in ℜ3 . But a recent algorithm of Garg =-=[25]-=-— discovered independently of our paper — works in any metric space. Geometric versions of polynomial time problems have been studied for many years, especially MST and Euclidean Matching. Exploiting ... |

75 | The Steiner problem with edge lengths 1 and 2
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Citation Context ... A spate of research activity in recent years (starting with the work of Zelikovsky[64]) has provided better algorithms, with an approximation ratio around 1.143 [65]. The metric case is MAX-SNP-hard =-=[13]-=-. k-TSP:. Given n nodes in ℜ d and an integer k>1, find the shortest tour that visits at least k nodes. An approximation algorithm due to Mata and Mitchell [46] K ≥ 8) is PLS-complete [41]. This stron... |

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63 |
Geometry Helps in Matching
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60 | S.-H.Teng. Parallel construction of quadtree and quality triangulations
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- 1993
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Citation Context ...roximation Schemes for Euclidean TSP and other Geometric Problems · 9 The shifted quadtree can be constructed easily using a sorting-based algorithm in O(n log 2 n) time; faster algorithms also exist =-=[12]-=-. step 3: dynamic programming Next we use dynamic programming to find the optimal (m, r)-light salesman path with respect to the shifted quadtree computed in Step 2, where m = O(c log n) and r = O(c).... |

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52 |
W.J.: Finding cuts in the TSP (A preliminary report
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(Show Context)
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- 1998
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Citation Context ...han Christofides’ for general metric spaces. Note that the results of [5] mentioned in the introduction seem to preclude a PTAS for general metrics. Recently Arora, Grigni, Karger, Klein and Woloszyn =-=[4]-=- extended our ideas and those of [30] to design a PTAS for any metric that is the shortestpath metric of a weighted planar graph (the paper [30] designed such PTASs for unweighted planar graphs). (ii)... |

49 |
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- 1995
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43 | When Hamming meets Euclid: the approximability of geometric TSP and MST
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42 |
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- 1980
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Citation Context ... recently proved by Du and Hwang [19]. A spate of research activity in recent years (starting with the work of Zelikovsky[64]) has provided better algorithms, with an approximation ratio around 1.143 =-=[65]-=-. The metric case is MAX-SNP-hard [13]. k-TSP:. Given n nodes in ℜ d and an integer k>1, find the shortest tour that visits at least k nodes. An approximation algorithm due to Mata and Mitchell [46] K... |

34 |
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Citation Context ...e as far as a factor 2/ √ 3 from the optimum. Furthermore, the famous Gilbert-Pollak [26] conjecture said it can’t be any further from the optimum; this conjecture was recently proved by Du and Hwang =-=[19]-=-. A spate of research activity in recent years (starting with the work of Zelikovsky[64]) has provided better algorithms, with an approximation ratio around 1.143 [65]. The metric case is MAX-SNP-hard... |

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31 | Low-degree spanning trees of small weight - Khuller, Raghavachari, et al. - 1996 |

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Citation Context ... the optimum. Two decades of research failed to improve upon Christofides’ algorithm for metric TSP. The Held-Karp heuristic is conjectured to have an approximation ratio 4/3 (some results of Goemans =-=[27]-=- support this conjecture) but the best upperbound known is 3/2 (Wolsey [63] and Shmoys and Williamson [58]). Some researchers continued to hope that even a PTAS might exist. A PTAS or Polynomial-Time ... |

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