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Partitioning a Weighted Tree into Subtrees with Weights in a Given Range

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Abstract

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are given integers such that 0≤l≤u. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such a partition is called an (l,u)-partition. We deal with three problems to find an (l,u)-partition of a given graph: the minimum partition problem is to find an (l,u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l,u)-partition with a given number p of components. All these problems are NP-hard even for series-parallel graphs, but are solvable in linear time for paths. In this paper, we present the first polynomial-time algorithm to solve the three problems for arbitrary trees.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Aoba-yama 6-6-05, Sendai, 980-8579, Japan

    Takehiro Ito & Xiao Zhou

  2. School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda, 669-1337, Japan

    Takao Nishizeki

  3. Fraunhofer Institute for Industrial Mathematics, ITWM, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    Michael Schröder

  4. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Takeaki Uno

Authors
  1. Takehiro Ito
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  2. Takao Nishizeki
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  3. Michael Schröder
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  4. Takeaki Uno
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  5. Xiao Zhou
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Correspondence to Takehiro Ito.

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Ito, T., Nishizeki, T., Schröder, M. et al. Partitioning a Weighted Tree into Subtrees with Weights in a Given Range. Algorithmica 62, 823–841 (2012). https://doi.org/10.1007/s00453-010-9485-y

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  • DOI: https://doi.org/10.1007/s00453-010-9485-y

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