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.

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 Top-k overlapping densest subgraphs: approximation algorithms and computational complexity

2020 ◽  
Author(s):  
Riccardo Dondi ◽  
Mohammad Mehdi Hosseinzadeh ◽  
Giancarlo Mauri ◽  
Italo Zoppis
Keyword(s):  
Computational Complexity ◽  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Graph Mining ◽  
Central Problem ◽  
Np Hard ◽  
Dense Subgraph ◽  
Dense Subgraphs ◽  
Approximation Factor

Abstract A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put in the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. An approach introduced to find possible overlapping subgraphs is the problem. Given an integer $$k \ge 1$$ k ≥ 1 and a parameter $$\lambda > 0$$ λ > 0 , the goal of this problem is to find a set of k dense subgraphs that may share some vertices. The objective function to be maximized takes into account the density of the subgraphs, the parameter $$\lambda $$ λ and the distance between each pair of subgraphs in the solution. The problem has been shown to admit a $$\frac{1}{10}$$ 1 10 -factor approximation algorithm. Furthermore, the computational complexity of the problem has been left open. In this paper, we present contributions concerning the approximability and the computational complexity of the problem. For the approximability, we present approximation algorithms that improve the approximation factor to $$\frac{1}{2}$$ 1 2 , when k is smaller than the number of vertices in the graph, and to $$\frac{2}{3}$$ 2 3 , when k is a constant. For the computational complexity, we show that the problem is NP-hard even when $$k=3$$ k = 3 .

scholarly journals Balanced Partition of Minimum Spanning Trees

2002 ◽  
pp. 26-35 ◽  
Author(s):  
Mattias Andersson ◽  
Joachim Gudmundsson ◽  
Christos Levcopoulos ◽  
Giri Narasimhan
Keyword(s):  
Spanning Trees ◽  
Minimum Spanning Trees ◽  
Balanced Partition
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