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# Approximation Algorithms for 2-Source Minimum Routing Cost k-Tree Problems

• Yen Hung Chen
• Gwo-Liang Liao
• Chuan Yi Tang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4707)

## Abstract

In this paper, we investigate some k-tree problems of graphs with given two sources. Let G = (V,E,w) be an undirected graph with nonnegative edge lengths and two sources s 1, s 2 ∈ V. The first problem is the 2-source minimum routing cost k -tree (2-kMRCT) problem, in which we want to find a tree T = (V T ,E T ) spanning k vertices such that the total distance from all vertex in V T to the two sources is minimized, i.e., we want to minimize $$\sum_{v\in V_T} \{d_T(s_1,v)+ d_T(s_2,v)\}$$, in which d T (s,v) is the length of the path between s and v on T. The second problem is the 2-source bottleneck source routing cost k -tree (2-kBSRT) problem, in which the objective function is the maximum total distance from any source to all vertices in V T , i.e., $$\max_{s\in (s_1,s_2)} \{ \sum_{v\in V_T} d_T(s,v) \}$$. The third problem is the 2-source bottleneck vertex routing cost k -tree (2-kBVRT) problem, in which the objective function is the maximum total distance from any vertex in V T to the two sources , i.e., $$\max_{v\in V_T}\left\{ d_T(s_1,v)+d_T(s_2,v) \right\}$$. In this paper, we present polynomial time approximation schemes (PTASs) for the 2-kMRCT and 2-kBVRT problems. For the 2-kBSRT problem, we give a (2 + ε)-approximation algorithm for any ε> 0.

## Keywords

combinatorial optimization problem k-tree approximation algorithm polynomial time approximation scheme (PTAS)

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2007

## Authors and Affiliations

• Yen Hung Chen
• 1
• Gwo-Liang Liao
• 1
• Chuan Yi Tang
• 2
1. 1.Department of Information Science and Management Systems, National Taitung University, 684, Sec.1, Chunghua Rd., Taitung 950, TaiwanR.O.C.
2. 2.Department of Computer Science, National Tsing Hua University, Hsinchu 300, TaiwanR.O.C.

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