## Low Degree Spanning Trees Of Small Weight (1996)

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Citations: | 30 - 2 self |

### BibTeX

@MISC{Khuller96lowdegree,

author = {Samir Khuller and Balaji Raghavachari and Neal Young},

title = {Low Degree Spanning Trees Of Small Weight},

year = {1996}

}

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### Abstract

. Given n points in the plane, the degree-K spanning tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing low-weight degree-K spanning trees for K ? 2. It is shown that for an arbitrary collection of n points in the plane, there exists a spanning tree of degree three whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree four whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in O(n) time. The results are generalized to points in higher dimensions. It is shown that for any d 3, an arbitrary collection of points in ! d contains a spanning tree of degree three, whose weight is at most 5/3 times the weight of a minimum spanning tre...

### Citations

10931 | Computers and Intractability: A Guide to the Theory of NPcompleteness. W.H. Freeman and Company. [The notion of approximation algorithms are presented ©Encyclopedia of Life Support - Garey, Johnson - 1979 |

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Citation Context ... is used to bound the cost of the obtained solution. Examples include Christofides’ heuristic for the traveling salesperson problem [3], biconnectivity augmentation [8], approximate weighted matching =-=[11]-=-, prize-collecting traveling salesperson [2], and bounded-degree subgraphs which have low weight and small bottleneck cost [16]. A question of general interest is how to obtain improved approximation ... |

273 |
Worst-Case Analysis of a New Heuristic for the Traveling Salesman Problem
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Citation Context ...dimitriou and Vazirani [15] posed the problem of obtaining factors better than two for the Euclidean degree-K spanning tree problem. It should be noted that in the special case of K = 2, Christofides =-=[3]-=- gave a simple and elegant polynomial time approximation algorithm with an approximation ratio of 1.5 for computing a traveling salesperson tour for points satisfying the triangle inequality (points i... |

80 | paths in grid graphs
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Citation Context ...o be the Euclidean distance between them. This problem is referred to as the Euclidean degree-K spanning tree problem and is a generalization of the Hamilton Path problem which is known to be NP-hard =-=[10, 12]-=-. When K = 3, it was shown to be NP-hard by Papadimitriou and Vazirani [15], who conjectured that it is NP-hard for K = 4 as well. When K = 5, the problem can be solved in polynomial time [14]. This p... |

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Citation Context ...straints. A polynomial-time algorithm to find a spanning tree or a Steiner tree of a given subset of vertices in a graph, with degree at most one more than minimum was given by Fürer and Raghavachari =-=[9]-=-. This was extended to weighted graphs by Fischer [7]. He showed how to find minimum spanning trees whose degree is within a constant multiplicative factor plus an additive O(log n) of the optimal deg... |

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Citation Context ...lution. Examples include Christofides’ heuristic for the traveling salesperson problem [3], biconnectivity augmentation [8], approximate weighted matching [11], prize-collecting traveling salesperson =-=[2]-=-, and bounded-degree subgraphs which have low weight and small bottleneck cost [16]. A question of general interest is how to obtain improved approximation algorithms for such problems when the points... |

56 |
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- 1992
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Citation Context ...ard [10, 12]. When K = 3, it was shown to be NP-hard by Papadimitriou and Vazirani [15], who conjectured that it is NP-hard for K = 4 as well. When K = 5, the problem can be solved in polynomial time =-=[14]-=-. This paper addresses the problem of computing low weight degree-K spanning trees for K > 2. In any metric space, it is known that there always exists a spanning tree of degree 2 whose cost is at mos... |

52 | Many birds with one stone: Multiobjective approximation algorithms
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Citation Context ...path Pv. The above technique of “shortcutting” the children of a vertex by “stringing” them together has been known before, especially in the context of computing degree-3 trees in metric spaces (see =-=[16, 18]-=-). Tree-3(V, T) — Find a degree 3 tree of V . 1 Root the MST T at a leaf vertex r. 2 For each vertex v ∈ V do 3 Compute Pv, the shortest path starting at v and visiting all the children of v. 4 Return... |

49 |
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Citation Context ...clidean degree-K spanning tree problem and is a generalization of the Hamilton Path problem which is known to be NP-hard [10, 12]. When K = 3, it was shown to be NP-hard by Papadimitriou and Vazirani =-=[15]-=-, who conjectured that it is NP-hard for K = 4 as well. When K = 5, the problem can be solved in polynomial time [14]. This paper addresses the problem of computing low weight degree-K spanning trees ... |

36 |
A proof of Gilbert-Pollak's conjecture on the Steiner ratio, Algorithmica 7
- Du, Hwang
- 1992
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Citation Context ...is requires making use of more than just the triangle inequality. Surprisingly, for most problems, improved algorithms are not known. (A notable exception is the famous Euclidean Steiner tree problem =-=[5, 6]-=-.) We use rudimentary geometric techniques to obtain an improved algorithm for the Euclidean degree-K spanning tree problem. The key to our method is to give short-cutting steps that are provably bett... |

35 |
On the relationship between the biconnectivity augmentation and traveling salesman problem
- Fredrickson, Jájá
- 1982
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Citation Context ... step where the triangle inequality is used to bound the cost of the obtained solution. Examples include Christofides’ heuristic for the traveling salesperson problem [3], biconnectivity augmentation =-=[8]-=-, approximate weighted matching [11], prize-collecting traveling salesperson [2], and bounded-degree subgraphs which have low weight and small bottleneck cost [16]. A question of general interest is h... |

28 |
On better heuristics for Euclidean Steiner minimum trees
- Du, Zhang, et al.
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Citation Context ...is requires making use of more than just the triangle inequality. Surprisingly, for most problems, improved algorithms are not known. (A notable exception is the famous Euclidean Steiner tree problem =-=[5, 6]-=-.) We use rudimentary geometric techniques to obtain an improved algorithm for the Euclidean degree-K spanning tree problem. The key to our method is to give short-cutting steps that are provably bett... |

19 | Constructing degree-3 spanners with other sparseness properties
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- 1996
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Citation Context ... a graph whose edge weights satisfy the triangle inequality. They give efficient algorithms for computing subgraphs which have low weight and small bottleneck cost. Salowe [18], and Das and Heffernan =-=[4]-=- consider the problem of computing bounded-degree graph spanners and provide algorithms for computing them. Robins and Salowe [17] study the maximum degrees of minimum spanning trees under various met... |

17 |
Optimizing the degree of minimum weight spanning trees
- Fischer
- 1485
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Citation Context ...ing tree or a Steiner tree of a given subset of vertices in a graph, with degree at most one more than minimum was given by Fürer and Raghavachari [9]. This was extended to weighted graphs by Fischer =-=[7]-=-. He showed how to find minimum spanning trees whose degree is within a constant multiplicative factor plus an additive O(log n) of the optimal degree. The degree bound is improved further in the case... |

13 | On the maximum degree of minimum spanning trees
- Robins, Salowe
- 1994
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Citation Context ...w weight and small bottleneck cost. Salowe [18], and Das and Heffernan [4] consider the problem of computing bounded-degree graph spanners and provide algorithms for computing them. Robins and Salowe =-=[17]-=- study the maximum degrees of minimum spanning trees under various metrics. 22. Preliminaries. Let V = {v1, . . .,vn} be a set of n points in the plane. Let G be the complete graph induced by V , whe... |

11 |
Euclidean spanner graphs with degree four
- Salowe
- 1994
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Citation Context ...n connectivity properties in a graph whose edge weights satisfy the triangle inequality. They give efficient algorithms for computing subgraphs which have low weight and small bottleneck cost. Salowe =-=[18]-=-, and Das and Heffernan [4] consider the problem of computing bounded-degree graph spanners and provide algorithms for computing them. Robins and Salowe [17] study the maximum degrees of minimum spann... |

2 |
Some extremal properties of convex sets
- Lillington
- 1975
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Citation Context ...is case, the right-hand side of (4) is exactly 3 √ 3. Thus, it suffices to show that the maximum perimeter achieved by any triangle whose vertices lie on a unit circle is 3 √ 3. This is easily proved =-=[13]-=-. Note that in an arbitrary metric space it is possible to have an (equilateral) triangle of perimeter six and a point X at distance one from each vertex. 3.2. Spanning trees of degree three. We now a... |