Euclidean Bounded-Degree Spanning Tree Ratios
نویسندگان
چکیده
منابع مشابه
Degree Bounded Spanning Trees
In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set S ⊆ V(G) of cardinality n(k − 1) + c + 2, there exists a vertex set X ⊆ S of cardinality k such that the degree sum of vertices in X is at least |V(G)| − c − 1. Then G has a spanning tree T with maximum de...
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* to be exact, times the weight of a minimum spanning tree (MST). In particular, we present an improved analysis of Chan’s degree-4 MST algorithm [4]. Previous results. Arora [1] and Mitchell [9] presented PTASs for TSP in Euclidean metric, for fixed dimensions. Unfortunately, neither algorithm extends to find degree-3 or degree-4 trees. Recently, Arora and Chang [3] have devised a quasi-polyno...
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Dirac’s classic theorem asserts that if G is a graph on n vertices, and δ(G) ≥ n/2, then G has a hamilton cycle. As is well known, the proof also shows that if deg(x) + deg(y) ≥ (n− 1), for every pair x, y of independent vertices in G, then G has a hamilton path. More generally, S. Win has shown that if k ≥ 2, G is connected and ∑ x∈I deg(x) ≥ n− 1 whenever I is a k-element independent set, the...
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This problem cab be shown to be NP-Hard by reducing Hamiltonian path to it. Essentially existence of a spanning tree of max degree two is equivalent to a having a Hamiltonian path in the graph. This also shows that the best approximation we can hope for (unless P = NP) is ∆∗ + 1 where ∆∗ is the max degree of the optimal tree. Note that we usually express approximations in a multiplicative form,...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2004
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-004-1117-3