The realization problem for Euclidean minimum spanning trees is NP-hard

1994 ◽  
Author(s):  
Peter Eades ◽  
Sue Whitesides
Keyword(s):  
Spanning Trees ◽  
Minimum Spanning Trees ◽  
Np Hard ◽  
Realization Problem ◽  
Euclidean Minimum Spanning Trees

scholarly journals Euclidean minimum spanning trees and bichromatic closest pairs

1991 ◽  
Vol 6 (3) ◽  
pp. 407-422 ◽  
Author(s):  
Pankaj K. Agarwal ◽  
Herbert Edelsbrunner ◽  
Otfried Schwarzkopf ◽  
Emo Welzl
Keyword(s):  
Spanning Trees ◽  
Minimum Spanning Trees ◽  
Euclidean Minimum Spanning Trees

scholarly journals SINGLE-SOURCE DILATION-BOUNDED MINIMUM SPANNING TREES

2013 ◽  
Vol 23 (03) ◽  
pp. 159-170
Author(s):  
OTFRIED CHEONG ◽  
CHANGRYEOL LEE
Keyword(s):  
Approximation Algorithm ◽  
Spanning Trees ◽  
Edge Length ◽  
Minimum Spanning Trees ◽  
Np Hard ◽  
Single Source ◽  
Vertex Set ◽  
Dilation Factor

Given a set S of points in the plane, a geometric network for S is a graph G with vertex set S and straight edges. We consider a broadcasting situation, where one point r ∊ S is a designated source. Given a dilation factor δ, we ask for a geometric network G such that for every point v ∊ S there is a path from r to v in G of length at most δ|rv|, and such that the total edge length is minimized. We show that finding such a network of minimum total edge length is NP-hard, and give an approximation algorithm.

scholarly journals BALANCED PARTITION OF MINIMUM SPANNING TREES

2003 ◽  
Vol 13 (04) ◽  
pp. 303-316 ◽  
Author(s):  
MATTIAS ANDERSSON ◽  
JOACHIM GUDMUNDSSON ◽  
CHRISTOS LEVCOPOULOS ◽  
GIRI NARASIMHAN
Keyword(s):  
Approximation Algorithm ◽  
Objective Function ◽  
Manufacturing Industry ◽  
Spanning Trees ◽  
Minimum Spanning Trees ◽  
Np Hard ◽  
Shipbuilding Industry ◽  
Specific Objective ◽  
Fixed Constant ◽  
Balanced Partition

To better handle situations where additional resources are available to carry out a task, many problems from the manufacturing industry involve dividing a task into a number of smaller tasks, while optimizing a specific objective function. In this paper we consider the problem of partitioning a given set [Formula: see text] of n points in the plane into k subsets, [Formula: see text], such that [Formula: see text] is minimized. Variants of this problem arise in applications from the shipbuilding industry. We show that this problem is NP-hard, and we also present an approximation algorithm for the problem, in the case when k is a fixed constant. The approximation algorithm runs in time O(n log n) and produces a partition that is within a factor (4/3+ε) of the optimal if k=2, and a factor (2+ε) otherwise.

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