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Capacitated Domination: Problem Complexity and Approximation Algorithms

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Abstract

We consider a local service-requirement assignment problem named capacitated domination from an algorithmic point of view. In this problem, we are given a graph with three parameters defined on each vertex, which are the cost, the capacity, and the demand, of a vertex, respectively. A vertex can be chosen multiple times in order to generate sufficient capacity for the demands of the vertices in its closed neighborhood. The objective of this problem is to compute a demand assignment of minimum cost such that the demand of each vertex is fully-served by some of its closed neighbors without exceeding the amount of capacity they provide.

In this paper, we provide complexity results as well as several approximation algorithms to compose a comprehensive study for this problem. First, we provide logarithmic approximations for general graphs which are asymptotically optimal. From the perspective of parameterized complexity, we show that this problem is W[1]-hard with respect to treewidth and solution size. Moreover, we show that this problem is fixed-parameter tractable with respect to treewidth and the maximum capacity of the vertices. The latter result implies a pseudo-polynomial time approximation scheme for planar graphs under a standard framework.

In order to drop the pseudo-polynomial factor, we develop a constant-factor approximation for planar graphs, based on a new perspective which we call general ladders on the hierarchical structure of outer-planar graphs. We believe that the approach we use can be applicable to other capacitated covering problems.

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Acknowledgements

The authors would like to thank the anonymous referees for their very helpful comments on the presentation of this work.

This work was supported in part by the National Science Council, Taipei 10622, Taiwan, under Grants NSC98-2221-E-001-007-MY3, NSC98-2221-E-001-008-MY3, NSC99-2911-I-002-055-2, NSC101-2221-E-005-026-MY2, and NSC101-2221-E-005-019-MY2.

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Author notes

  1. D. T. Lee

    Present address: Department of Computer Science and Engineering, National Chung-Hsing University, Taichung, Taiwan

Authors and Affiliations

  1. Institute of Information Science, Academia Sinica, Taipei, Taiwan

    Mong-Jen Kao & D. T. Lee

  2. Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan

    Han-Lin Chen

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  1. Mong-Jen Kao
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Correspondence to Mong-Jen Kao.

Additional information

Extended abstracts of this work appeared in the 4th Frontiers of Algorithmics Workshop (FAW’10), Wuhan, China (received the best student paper award) [31] and the 22nd International Symposium on Algorithms and Computation (ISAAC’11), Yokohama, Japan [32].

Part of this work was done when the author M.-J. Kao was with Karlsruhe Institute of Technology (KIT), Germany, as a visiting student.

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Kao, MJ., Chen, HL. & Lee, D.T. Capacitated Domination: Problem Complexity and Approximation Algorithms. Algorithmica 72, 1–43 (2015). https://doi.org/10.1007/s00453-013-9844-6

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