Approximation algorithm for the balanced 2-connected k-partition problem

For two positive integers m,k and a connected graph G=(V,E) with a nonnegative vertex weight function w, the balanced m-connected k-partition problem, denoted as BCmPk, is to find a partition of V into k disjoint nonempty vertex subsets (V1,V2,…,Vk) such that each G[Vi] (the subgraph of G induced by...

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Bibliographic Details
Published in:Theoretical computer science Vol. 609; pp. 627 - 638
Main Authors: Wu, Di, Zhang, Zhao, Wu, Weili
Format: Journal Article
Language:English
Published: Elsevier B.V 04.01.2016
Subjects:
Algorithms
Approximation
Balanced m-connected k-partition
Balancing
FPTAS
Graphs
Interval graph
Intervals
Lower bounds
Mathematical analysis
Pseudo-polynomial time algorithm
Weight reduction
Interval graph
Balanced m-connected k-partition
FPTAS
Pseudo-polynomial time algorithm
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary:For two positive integers m,k and a connected graph G=(V,E) with a nonnegative vertex weight function w, the balanced m-connected k-partition problem, denoted as BCmPk, is to find a partition of V into k disjoint nonempty vertex subsets (V1,V2,…,Vk) such that each G[Vi] (the subgraph of G induced by Vi) is m-connected, and min1≤i≤k⁡{w(Vi)} is maximized. The optimal value of BCmPk on graph G is denoted as βm⁎(G,k), that is, βm⁎(G,k)=max⁡min1≤i≤k⁡{w(Vi)}, where the maximum is taken over all m-connected k-partition of G. In this paper, we study the BC2Pk problem on interval graphs, and obtain the following results. (1) For k=2, a 4/3-approximation algorithm is given for BC2P2 on 4-connected interval graphs. (2) In the case that there exists a vertex v with weight at least W/k, where W is the total weight of the graph, we prove that the BC2Pk problem on a 2k-connected interval graph G can be reduced to the BC2Pk−1 problem on the (2k−1)-connected interval graph G−v. In the case that every vertex has weight at most W/k, we prove a lower bound β2⁎(G,k)≥W/(2k−1) for 2k-connected interval graph G. (3) Assuming that weight w is integral, a pseudo-polynomial time algorithm is obtained. Combining this pseudo-polynomial time algorithm with the above lower bound, a fully polynomial time approximation scheme (FPTAS) is obtained for the BC2Pk problem on 2k-connected interval graphs.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2015.02.001