## Approximation schemes for NP-hard geometric optimization problems: A survey (2003)

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Venue: | Mathematical Programming |

Citations: | 38 - 2 self |

### BibTeX

@ARTICLE{Arora03approximationschemes,

author = {Sanjeev Arora},

title = {Approximation schemes for NP-hard geometric optimization problems: A survey},

journal = {Mathematical Programming},

year = {2003},

volume = {97},

pages = {2003}

}

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### Abstract

NP-hard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NP-hard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), k-TSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), k-MST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum

### Citations

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Citation Context ...omial-time algorithm that approximates the problem within a factor 1 + ɛ. Context. Designing approximation algorithms for NP-hard problems is a well-developed science; see the books by Hochbaum (ed.) =-=[34]-=- and Vazirani [63]. The most popular method involves solving a mathematical programming relaxation (either a linear or semidefinite program) and rounding the fractional solution thus obtained to an in... |

572 |
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Citation Context ...endence on dimension seems consistent with known complexity results. Trevisan [60] has shown that the Euclidean TSP problem becomes MAX-SNP-hard in O(log n) dimensions, which means —by the results of =-=[53, 8]-=-— that there is a γ > 1 such that approximation within a factor 1 + γ is NP-hard. 1 Introduction to the TSP algorithm In this section we describe a PTAS for TSP in ℜ 2 with ℓ2 norm; the generalization... |

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Citation Context ...ce Trevisan [60] has shown that there is an ɛ > 0 such that (1 + ɛ)approximation to Euclidean TSP in n dimensions is MAX-SNP hard. Using a general dimension-reduction due to Johnson and Lindenstrauss =-=[37]-=-, it follows that if a polynomial-time approximation scheme exists even in 14O(log n) dimensions, then P = NP. 5 Generalizing to other problems: a methodology The design of the PTAS for the TSP uses ... |

389 | A Separator Theorem for Planar Graphs
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Citation Context ...n “almost local” quality (see Section 2 for an example). Such a Structure theorem appears implicitly in descriptions of most earlier PTASs, including the ones for Knapsack [35], planar graph problems =-=[45, 46, 11, 33, 40]-=-, and most recently, for scheduling to minimize average completion time [2, 39]. These PTASs involve a simple divide-and-conquer approach or dynamic programming to optimize over the set of “almost loc... |

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Citation Context ...n “almost local” quality (see Section 2 for an example). Such a Structure theorem appears implicitly in descriptions of most earlier PTASs, including the ones for Knapsack [35], planar graph problems =-=[45, 46, 11, 33, 40]-=-, and most recently, for scheduling to minimize average completion time [2, 39]. These PTASs involve a simple divide-and-conquer approach or dynamic programming to optimize over the set of “almost loc... |

278 | P-complete approximation problems - Sahni, Gonzales - 1976 |

273 |
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Citation Context ...or. (One of the exceptions is k-median, for which no constant factor approximation was known at the time a PTAS was found [9].) For the TSP, the best previous algorithm was the Christofides heuristic =-=[20]-=-, which approximates the problem within a factor 1.5 in polynomial time. The decision version of Euclidean TSP (“Does a tour of cost ≤ C exist?”) is NP-hard [51, 28], but is not known to be in NP beca... |

271 |
The travelling salesman problem: A case study in local optimization
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Citation Context ...to Euclidean TSP in constant dimensions. 2gorithm of this survey, though its asymptotic running time is nearly linear, is not competitive with existing implementations of other TSP heuristics (e.g., =-=[36, 4]-=-). But maybe our algorithms for other geometric problems will be more competitive. One of the goals of the current survey is to serve as a tutorial for the reader who wishes to apply the techniques de... |

259 | TSPLIB – A Traveling Salesman Problem Library - Reinelt - 1991 |

187 |
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Citation Context ...roximate solution that has an “almost local” quality (see Section 2 for an example). Such a Structure theorem appears implicitly in descriptions of most earlier PTASs, including the ones for Knapsack =-=[35]-=-, planar graph problems [45, 46, 11, 33, 40], and most recently, for scheduling to minimize average completion time [2, 39]. These PTASs involve a simple divide-and-conquer approach or dynamic program... |

185 | Approximating clique is almost NPcomplete - Feige, Goldwasser, et al. - 1991 |

174 |
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Citation Context ...(n/poly(ɛ)) edges in which the distance between any pair of nodes is within a factor (1 + ɛ) of the Euclidean distance. Such a spanner can be computed in O(n log n/poly(ɛ)) time (see Althoefer et al. =-=[3]-=-). Note that distances in the spanner define a metric space in which the optimum TSP cost is within a factor (1 + ɛ) of the optimum in the Euclidean space. Rao and Smith notice that the tour transform... |

164 | Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems
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- 1999
(Show Context)
Citation Context ..., can decide if ∑ √ i ai ≤ C. ) Arora’s paper gave the first PTASs for many of these problems in 1996. A few months later Mitchell independently discovered a similar nO(1/ɛ) time approximation scheme =-=[50]-=-; this algorithm used ideas from the earlier paper of Mata and Mitchell [47]. The running time of Arora’s and Mitchell’s algorithms was nO(1/ɛ) , but Arora later improved the running time of his algor... |

163 |
Applications of a Planar Separator Theorem
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- 1980
(Show Context)
Citation Context ...n “almost local” quality (see Section 2 for an example). Such a Structure theorem appears implicitly in descriptions of most earlier PTASs, including the ones for Knapsack [35], planar graph problems =-=[45, 46, 11, 33, 40]-=-, and most recently, for scheduling to minimize average completion time [2, 39]. These PTASs involve a simple divide-and-conquer approach or dynamic programming to optimize over the set of “almost loc... |

152 |
The Shortest Path Through Many Points
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Citation Context ...impler analysis in Section 2— reason in an edge-by-edge fashion. We will use the following well-known fact about Euclidean TSP that is implicit in the analysis of Karp’s dissection heuristic (or even =-=[13]-=-), and is made explicit in [5]. Lemma 4 (Patching Lemma) Let S be any line segment of length s and π be a closed path that crosses S at least thrice. Then we can break the path in all but two of these... |

140 |
Steiner minimal trees
- Gilbert, Pollak
- 1968
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Citation Context ...riangle in ℜ2 (with distances measured in ℓ2 norm), the optimum Steiner tree contains the centroid of the triangle, and has cost √ 3/2 factor lower than the MST. Furthermore, the famous GilbertPollak =-=[29]-=- conjecture said that for every set of input nodes, a Steiner tree has cost at least √ 3/2 times the cost of the MST. Du and Hwang [23] proved this conjecture and thus showed that the MST is a 2/ √ 3-... |

114 | Approximation schemes for euclidean k-medians and related problems
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- 1998
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Citation Context ...pproximation algorithms that approximate the problem within some constant factor. (One of the exceptions is k-median, for which no constant factor approximation was known at the time a PTAS was found =-=[9]-=-.) For the TSP, the best previous algorithm was the Christofides heuristic [20], which approximates the problem within a factor 1.5 in polynomial time. The decision version of Euclidean TSP (“Does a t... |

103 |
An 11/6-approximation algorithm for the network Steiner problem, Algorithmica 9
- Zelikovsky
- 1993
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Citation Context ... Hwang [23] proved this conjecture and thus showed that the MST is a 2/ √ 3-approximation to the optimum Steiner tree. A spate of research activity in recent years starting with the work of Zelikovsky=-=[65]-=- has provided better approximation algorithms, with an approximation ratio around 1.143 [66]. The metric case does not have an approximation scheme if P ≠ NP [16]. The Steiner Tree problem involves an... |

94 |
Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems
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Citation Context ...eason in an edge-by-edge fashion. We will use the following well-known fact about Euclidean TSP that is implicit in the analysis of Karp’s dissection heuristic (or even [13]), and is made explicit in =-=[5]-=-. Lemma 4 (Patching Lemma) Let S be any line segment of length s and π be a closed path that crosses S at least thrice. Then we can break the path in all but two of these places, and add to it line se... |

91 |
Probabilistic Analysis of Partitioning Algorithms for the Traveling Salesman Problem in the Plane
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- 1977
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Citation Context ...t of geometric algorithms, divide-and-conquer ideas are quite old, though they had not resulted in PTASs until recently. Specifically, geometric divide and conquer appears Karp’s dissection heuristic =-=[38]-=-, Smith’s 2O(√n) time exact algorithm for TSP [59], and Blum, Chalasani and Vempala’s approximation algorithm for k-MST [19], and Mata and Mitchell’s constantfactor approximations for many geometric p... |

85 | Some NP-complete geometric problems
- Garey, Graham, et al.
- 1976
(Show Context)
Citation Context ...gorithm was the Christofides heuristic [20], which approximates the problem within a factor 1.5 in polynomial time. The decision version of Euclidean TSP (“Does a tour of cost ≤ C exist?”) is NP-hard =-=[51, 28]-=-, but is not known to be in NP because of the use of square roots in computing the edge costs2 . Specifically, there is no known polynomial-time algorithm that, given integers a1, a2, . . . , an, C, c... |

85 | An improved approximation ratio for the minimum latency problem
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- 1996
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Citation Context ... metric instances. Blum, Chalasani, Coppersmith, Pulleyblank, Raghavan and Sudan gave a 144-approximation algorithm for the metric case and a 8-approximation for weighted trees. Goemans and Kleinberg =-=[31]-=- then gave a 21.55-approximation in the metric case and a 3.59..-approximation in the geometric case (the latter uses the PTAS for k-TSP). Arora and Karakostas [7] designed a quasipolynomial-time appr... |

82 | Finding Small Simple Cycle Separators for 2-Connected Planar Graphs - Miller - 1986 |

78 | Approximation algorithms for geometric problems
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- 1996
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Citation Context ...he optimum solution for I. (The preceding definition is for minimization problems; for maximization problems α ≤ 1.) Sometimes we use the shortened name “α-approximation algorithm.” Bern and Eppstein =-=[14]-=- give an excellent survey circa 1995 of approximation algorithms for geometric problems. The current survey will concentrate on developments subsequent to 1995, many of which followed the author’s dis... |

72 | The Steiner problem with edge lengths 1 and 2
- Bern, Plassman
- 1989
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Citation Context ...s starting with the work of Zelikovsky[65] has provided better approximation algorithms, with an approximation ratio around 1.143 [66]. The metric case does not have an approximation scheme if P ≠ NP =-=[16]-=-. The Steiner Tree problem involves an objective function that is a sum of edge lengths and it obeys the Patching Lemma (as is easily checked). Now we briefly describe the algorithm. First we perturb ... |

67 | Approximation algorithms for geometric tour and network design problems
- Mata, Mitchell
- 1995
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Citation Context ...s 2O(√n) time exact algorithm for TSP [59], and Blum, Chalasani and Vempala’s approximation algorithm for k-MST [19], and Mata and Mitchell’s constantfactor approximations for many geometric problems =-=[47]-=-. Surprisingly, the proofs of the structure theorems for geometric problems are elementary and this survey will describe them essentially completely. We also survey a more recent result of Rao and Smi... |

61 | Approximating geometrical graphs via spanners and banyans
- Rao, Smith
- 1998
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Citation Context ...urprisingly, the proofs of the structure theorems for geometric problems are elementary and this survey will describe them essentially completely. We also survey a more recent result of Rao and Smith =-=[55]-=- that improves the running time for some problems. We will be concerned only with asymptotics and hence not with practical implementations. The TSP al1 In fact, the discovery of this algorithm stemmed... |

59 | Parallel construction of quadtrees and quality triangulations
- Bern, Eppstein, et al.
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Citation Context ...s just the quadtree of the enclosing box) can be efficiently computed, for instance by sorting the nodes by xand y-coordinates (for better algorithms, especially in higher dimensions, see Bern et al. =-=[15]-=-). The dynamic programming now is the obvious one. Suppose we are interested in portal-respecting tours that enter/exit each dissection square at most 4k times. The subproblem inside the square can be... |

55 | Geometry helps in matching - Vaidya - 1989 |

49 | A.: A polynomial-time approximation scheme for weighted planar graph TSP - Arora, Grigni, et al. - 1998 |

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

48 | Analyzing the HeldKarp TSP bound: a monotonicity property with application - Shmoys, Williamson - 1990 |

43 | When Hamming Meets Euclid: The Approximability of Geometric TSP and MST
- Trevisan
- 1997
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Citation Context ...he Real RAM model. 3then the algorithm takes superpolynomial time that grows to exponential around d = O(log n). This dependence on dimension seems consistent with known complexity results. Trevisan =-=[60]-=- has shown that the Euclidean TSP problem becomes MAX-SNP-hard in O(log n) dimensions, which means —by the results of [53, 8]— that there is a γ > 1 such that approximation within a factor 1 + γ is NP... |

43 | Better approximation bounds for the network and Euclidean Steiner tree problems
- Zelikovsky
- 1996
(Show Context)
Citation Context ... the optimum Steiner tree. A spate of research activity in recent years starting with the work of Zelikovsky[65] has provided better approximation algorithms, with an approximation ratio around 1.143 =-=[66]-=-. The metric case does not have an approximation scheme if P ≠ NP [16]. The Steiner Tree problem involves an objective function that is a sum of edge lengths and it obeys the Patching Lemma (as is eas... |

41 | Heuristic analysis, linear programming and branch and bound,” in Combinatorial Optimization II, ser. Mathematical Programming Studies - Wolsey - 1980 |

36 |
A proof of Gilbert-Pollak's conjecture on the Steiner ratio, Algorithmica 7
- Du, Hwang
- 1992
(Show Context)
Citation Context ...factor lower than the MST. Furthermore, the famous GilbertPollak [29] conjecture said that for every set of input nodes, a Steiner tree has cost at least √ 3/2 times the cost of the MST. Du and Hwang =-=[23]-=- proved this conjecture and thus showed that the MST is a 2/ √ 3-approximation to the optimum Steiner tree. A spate of research activity in recent years starting with the work of Zelikovsky[65] has pr... |

35 | Towards a syntactic characterization of PTAS
- Khanna, Motwani
- 1996
(Show Context)
Citation Context |

32 | A constant factor approximation for the k-mst problem
- Blum, Ravi, et al.
- 1996
(Show Context)
Citation Context ...fically, geometric divide and conquer appears Karp’s dissection heuristic [38], Smith’s 2 O(√ n) time exact algorithm for TSP [59], and Blum, Chalasani and Vempala’s approximation algorithm for k-MST =-=[19]-=-, and Mata and Mitchell’s constantfactor approximations for many geometric problems [47]. Surprisingly, the proofs of the structure theorems for geometric problems are elementary and this survey will ... |

30 | Approximation schemes for minimum latency problems
- Arora, Karakostas
- 1999
(Show Context)
Citation Context ...ighted trees. Goemans and Kleinberg [31] then gave a 21.55-approximation in the metric case and a 3.59..-approximation in the geometric case (the latter uses the PTAS for k-TSP). Arora and Karakostas =-=[7]-=- designed a quasipolynomial-time approximation scheme for the problem. We do not know whether the running time can be reduced to polynomial. Consider the objective function for the problem: we have to... |

30 | Low degree spanning trees of small weight
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- 1994
(Show Context)
Citation Context ...path problem is a subcase when d = 2. Every minimum spanning tree has degree at most 5, so the problem is trivial for d ≥ 5. The case d = 4 is NPhard and the status when d = 3 is open. Khuller et al. =-=[41]-=- give 1.5and 1.25-approximations for the two problems. The techniques of Section 2 seem very applicable but there is as yet no PTAS. (The author has certainly has tried to design one, and maybe others... |

29 | A randomized approximation scheme for metric max-cut
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- 1998
(Show Context)
Citation Context ...uclidean Max-Cut Given n nodes, find a partition into two subsets S1, S2 that maximises the sum of the lengths of the edges that have an endpoint in each of S1 and S2. Fernandez de la Vega and Kenyon =-=[22]-=- give a PTAS for this problem. The techniques are unrelated to those covered in our survey and also extend to any metric space. Maximum traveling salesman The maximization version of the usual TSP —fi... |

25 | Worst-case Comparison of Valid Inequalities for the TSP, mathematical Programming 69 - Goemans - 1995 |

23 |
Euclidean TSP is NP-complete
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(Show Context)
Citation Context ...gorithm was the Christofides heuristic [20], which approximates the problem within a factor 1.5 in polynomial time. The decision version of Euclidean TSP (“Does a tour of cost ≤ C exist?”) is NP-hard =-=[51, 28]-=-, but is not known to be in NP because of the use of square roots in computing the edge costs2 . Specifically, there is no known polynomial-time algorithm that, given integers a1, a2, . . . , an, C, c... |

20 |
The complexity of the travelling repairman problem. Informatique Theoretique et Applications
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(Show Context)
Citation Context ...e for the interface and other tricks, the running time can be made polynomial [9] and even near linear [42]. 5.3 Minimum Latency The minimum latency problem, also known as traveling repairman problem =-=[1]-=-, is a variant of the TSP in which the starting node of the tour is given and the goal is to minimize the sum of the arrival times at the other nodes. (The arrival time is the distance covered before ... |

20 | The complexity of the Lin-Kernighan heuristic for the traveling salesman problem - Papadimitriou - 1992 |

12 |
On the solution of traveling salesman problems,” Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung
- 37Applegate, Bixby, et al.
- 1998
(Show Context)
Citation Context ...to Euclidean TSP in constant dimensions. 2sgorithm of this survey, though its asymptotic running time is nearly linear, is not competitive with existing implementations of other TSP heuristics (e.g., =-=[36, 4]-=-). But maybe our algorithms for other geometric problems will be more competitive. One of the goals of the current survey is to serve as a tutorial for the reader who wishes to apply the techniques de... |