## 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: | 325 - 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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Citation Context ...or a fascinating history, see Lawler et al. [43]. Since the 1970s, mounting evidence from complexity theory suggests that the problem is computationally difficult. Exact optimization 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 [... |

693 | Approximation algorithms for combinatorial problems - Johnson - 1974 |

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577 |
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Citation Context ...wani, Sudan, and Szegedy [5] showed that if P ̸= NP, then metric TSP and many other problems do not have a PTAS. Their work relied upon the theory of MAX-SNPcompleteness (Papadimitriou and Yannakakis =-=[53]-=-), the notion of probabilistically checkable proofs or PCPs (Feige, Goldwasser, Lovász, Safra and Szegedy [22], Arora and Safra [7]), and the connection between PCPs and hardness of approximation [22]... |

417 | Extensions of lipschitz mappings into a hilbert space - Johnson, Lindenstrauss - 1982 |

373 | Probabilistic Checking of Proofs: A new characterization of NP
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Citation Context ...pon the theory of MAX-SNPcompleteness (Papadimitriou and Yannakakis [53]), the notion of probabilistically checkable proofs or PCPs (Feige, Goldwasser, Lovász, Safra and Szegedy [22], Arora and Safra =-=[7]-=-), and the connection between PCPs and hardness of approximation [22]. The status of Euclidean TSP remained open, however. In this paper, we show that Euclidean TSP has a PTAS. For every fixed c>1, a ... |

330 |
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Citation Context ...rograms. This may be especially true for problems such as Steiner Tree or k-MST, for which no good heuristics are known. For the TSP, classical local-exchange 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 ... |

327 | Bounds for certain multiprocessing anomalies - Graham |

321 |
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Citation Context ...lgorithmic ideas during the past few decades, and influenced the emergence of fields such as operations research, polyhedral theory and complexity theory. For a fascinating history, see Lawler et al. =-=[43]-=-. Since the 1970s, mounting evidence from complexity theory suggests that the problem is computationally difficult. Exact optimization is NP-hard (Karp [38]). So is approximating the optimum within an... |

282 |
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Citation Context ...rom complexity theory suggests that the problem is computationally difficult. Exact optimization 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... |

281 |
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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 ... |

280 |
The Traveling Salesman Problem: A Case Study
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Citation Context ...Tree or k-MST, for which no good heuristics are known. For the TSP, classical local-exchange 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, ef... |

271 |
TSPLIB—A Traveling Salesman Problem Library
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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... |

260 | Computer solutions to the travelling salesman problem - Lin - 1965 |

207 |
Proof verification and the intractability of approximation problems
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- 1992
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Citation Context ...ms; two important ones are Subset-Sum (Ibarra and Kim [32]) and Bin-Packing (Fernandez de la Vega and Lueker [18]; see also Karmarkar and Karp [37]). Recently Arora, Lund, Motwani, Sudan, and Szegedy =-=[5]-=- showed that if P ̸= NP, then metric TSP and many other problems do not have a PTAS. Their work relied upon the theory of MAX-SNPcompleteness (Papadimitriou and Yannakakis [53]), the notion of probabi... |

200 | How easy is local search
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- 1988
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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... |

187 |
Fast approximation algorithms for the knapsack and sum of subset problems
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Citation Context ...r 1 + 1/c. The running time could depend upon c, but for each fixed c has to be polynomial in the input size. PTAS’s are known for very few problems; two important ones are Subset-Sum (Ibarra and Kim =-=[32]-=-) and Bin-Packing (Fernandez de la Vega and Lueker [18]; see also Karmarkar and Karp [37]). Recently Arora, Lund, Motwani, Sudan, and Szegedy [5] showed that if P ̸= NP, then metric TSP and many other... |

185 |
Approximating clique is almost np-complete
- Feige, Goldwasser, et al.
- 1991
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Citation Context ...S. Their work relied upon the theory of MAX-SNPcompleteness (Papadimitriou and Yannakakis [53]), the notion of probabilistically checkable proofs or PCPs (Feige, Goldwasser, Lovász, Safra and Szegedy =-=[22]-=-, Arora and Safra [7]), and the connection between PCPs and hardness of approximation [22]. The status of Euclidean TSP remained open, however. In this paper, we show that Euclidean TSP has a PTAS. Fo... |

166 | Guillotine subdivisions approximate polygonal subdivisions: Part II- A simple PTAS for geometric k-MST, TSP, and related problems. Preliminary manuscript
- Mitchell
- 1996
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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... |

160 |
The shortest path through many points
- Beardwood, Halton, et al.
- 1959
(Show Context)
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... |

146 |
Steiner minimal trees
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- 1968
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Citation Context ...mum spanning tree is not an optimal solution. In ℜ 2 (with distances measured in ℓ2 norm) the cost of the MST can be 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[... |

135 |
Fast algorithms for geometric traveling salesman problem
- Bentley
- 1992
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Citation Context ...Tree or k-MST, for which no good heuristics are known. For the TSP, classical local-exchange 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, ef... |

121 |
An ecient approximation scheme for the one-dimensional binpacking problem
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- 1982
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Citation Context ...l in the input size. PTAS’s are known for very few problems; two important ones are Subset-Sum (Ibarra and Kim [32]) and Bin-Packing (Fernandez de la Vega and Lueker [18]; see also Karmarkar and Karp =-=[37]-=-). Recently Arora, Lund, Motwani, Sudan, and Szegedy [5] showed that if P ̸= NP, then metric TSP and many other problems do not have a PTAS. Their work relied upon the theory of MAX-SNPcompleteness (P... |

115 | Approximation schemes for the Euclidean k-medians and related problems
- Arora, Raghavan, et al.
- 1998
(Show Context)
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... |

103 |
An 11=6-approximation algorithm for the network Steiner Problem. Algorithmica
- Zelikovsky
- 1993
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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... |

98 |
Probabilistic analysis of partitioning algorithms for the traveling salesman problem in the plane
- Karp
- 1977
(Show Context)
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... |

95 |
Polynomial-time approximation schemes for Euclidean TSP and other geometric problems
- Arora
- 1996
(Show Context)
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 ... |

92 |
The complexity of facets (and some facets of complexity
- Papadimitriou, Yannakakis
- 1982
(Show Context)
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 (... |

86 | Some NP-complete geometric problems
- Garey, Graham, et al.
- 1976
(Show Context)
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... |

80 | Approximation algorithms for geometric problems
- Bern, Eppstein
(Show Context)
Citation Context ...tching, all are NP-hard. Prior to our work, the best approximation algorithms for the NP-hard problems achieved a constant factor approximation in polynomial time (see the survey by Bern and Eppstein =-=[11]-=-). These algorithms used problem-specific ideas and usually require at least Ω(n2 ) time (sometimes a lot more). In contrast, our approximation schemes for the different problems rely on essentially t... |

78 |
A 3-approximation for the minimum tree spanning k vertices
- Garg
- 1996
(Show Context)
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
- Bern, Plassmann
- 1989
(Show Context)
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... |

68 | Approximation Algorithms for Geometric tour and network problems
- Mata, Mitchell
- 1995
(Show Context)
Citation Context ...3 [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 strongly suggests that no polynomial-time algorithm can find such a local optimum; see [35]. Many variants of Lin-Kernighan are also PLS-complete [51]. 2 It appears ... |

61 | S.-H.Teng. Parallel construction of quadtree and quality triangulations
- Bern, Eppstein
- 1993
(Show Context)
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).... |

61 |
Geometry helps in matching
- Vaidya
- 1988
(Show Context)
Citation Context ...minimum cost set of nonadjacent edges that cover all vertices. This problem can be solved in polynomial time (even for nongeometric instances). Vaidya shows how to solve it optimally in Õ(n2.5 ) time =-=[60]-=-, and to approximate it within a factor (1 + 1/c) inO(poly(c)n 1.5 log 3 n) time [61]. Theorem 1. For each fixed d, the ℜ d version of each of the above problems has a randomized PTAS. The algorithm c... |

61 | Approximating geometric graphs via \spanners" and \banyans
- Rao, Smith
- 1998
(Show Context)
Citation Context ...>1 that finding a(1+1/c)-approximate solution in ℜ O(log n) is NP-hard. Trevisan’s work does not preclude the possibility of a running time such as O(n log 2 n +2 O(c) )inℜ 2 . Recently Rao and Smith =-=[54]-=- improved upon our ideas to achieve a running time of (c √ d) O(d(√ dc) d−1 ) n + O(d log n) inℜ d ; thus their running time in ℜ 2 is O(c O(c) n + n log n). More recently, Czumaj and Lingas [17] impr... |

57 | The Steiner problem with edge lengths 1 - Bern, Plassmann - 1989 |

56 | Iterated nearest neighbors and finding minimal polytopes
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- 1994
(Show Context)
Citation Context ...solution to k-TSP is at most 2 √ k times larger than the size of the smallest square containing at least k points (in ℜ d it is at most dk 1 1− d times the size of such a cube). Eppstein and Erickson =-=[21]-=- show how to approximate the size of this square within a factor 2 by computing for each of the n nodes its k nearest neighbors. This takes O(nk log n) time 5 . Once we have an approximation A such th... |

52 | A polynomial-time approximation scheme for weighted planar graph TSP
- Arora, Grigni, et al.
- 1998
(Show Context)
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)... |

50 |
Finding cuts in the TSP (A preliminary report
- Applegate, Bixby, et al.
(Show Context)
Citation Context ...s the one for the TSP. 4. HOW PRACTICAL IS OUR PTAS? Many years of research and rapid increase in computer power have led to linear programming methods — the best of which currently is branch-and-cut =-=[2]-=-— that in less than an hour find provably optimal tours on instances with a few hundred nodes. Finding optimal tours on a few thousand nodes takes longer, however. Furthermore, the convergence rate of... |

49 | An approximation scheme for planar graph TSP
- Grigni, Koutsoupias, et al.
- 1995
(Show Context)
Citation Context ...ty in 2 O(√ n) time; the main idea is that the plane can be recursively partitioned such that an optimal tour cross every partition only O( √ n) times. Recently Grigni, Koutsoupias, and Papadimitriou =-=[30]-=- designed an approximation scheme for planar graph TSP using similar ideas. Finally, we must address the inevitable question: Is our PTAS practical? A straightforward implementation (for even moderate... |

48 |
Analyzing the Held-Karp TSP bound: A monotonicity property wtih application
- Shmoys, Williamson
- 1990
(Show Context)
Citation Context ...eld-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 Approximation Scheme is a polynomial-time algorithm — or a family of such algorithms— that, for each fixed... |

43 | When Hamming meets Euclid: the approximability of geometric TSP and MST
- Trevisan
- 1997
(Show Context)
Citation Context ... that computes (1 + c)-approximations must have a running time with some exponential dependence on c (unless, of course, NP-complete problems can be solved in subexponential time). Likewise, Trevisan =-=[59]-=- has presented evidence that every approximation scheme for ℜ d must have a running time with a doubly exponential dependence on d. He shows for some fixed c>1 that finding a(1+1/c)-approximate soluti... |

43 | Better approximation bounds for the network and Euclidean Steiner tree problems
- Zelikovsky
- 1996
(Show Context)
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... |

42 |
Heuristic analysis, linear programming and branch and bound
- Wolsey
- 1980
(Show Context)
Citation Context ...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 Approximation Scheme is a polynomial-time algorithm — or a family of such a... |

36 |
A proof of Gilbert-Pollak's conjecture on the Steiner ratio
- Du, Hwang
- 1992
(Show Context)
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... |

32 | A constant-factor approximation for the k-MST problem in the plane
- Blum, Chalasani, et al.
- 1995
(Show Context)
Citation Context ...constant factor approximation in ℜ2 . k-MST:. Given n nodes in ℜd and an integer k ≥ 2, find k nodes with the shortest Minimum Spanning Tree. The problem is NP-hard [23]. Blum, Chalasani, and Vempala =-=[14]-=- gave the first O(1)-factor approximation algorithm for points in ℜ2 and Mitchell [48] improved this factor to 2 √ 2. Euclidean Min-Cost Perfect Matching:. (EMCPM) Given 2n points in ℜ 2 (or ℜ d in ge... |

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

28 | Weighted k-cardinality trees: complexity and polyhedral structure, Networks 24 - Fischetti, Hamacher, et al. - 1994 |

27 | Worst-case comparison of valid inequalities for the TSP
- Goemans
- 1995
(Show Context)
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 ... |