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Minmax Centered k-Partitioning of Trees and Applications to Sink Evacuation with Dynamic Confluent Flows

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Abstract

Let \(T=(V,E)\) be a tree with associated costs on its subtrees. A minmax k-partition of T is a partition into k subtrees, minimizing the maximum cost of a subtree over all possible partitions. In the centered version of the problem, the cost of a subtree is defined as the minimum cost of “servicing” that subtree using a center located within it. The problem motivating this work was the sink-evacuation problem on trees, i.e., finding a collection of k-sinks that minimize the time required by a confluent dynamic network flow to evacuate all supplies to sinks. This paper provides the first polynomial-time algorithm for solving this problem, running in \(O\Bigl ( \max (k \log k,\log n) k^2 n \log ^4 n\Bigr )\) time. The technique developed can be used to solve any Minmax Centered k-Partitioning problem on trees in which the servicing costs satisfy some very general conditions. Solutions can be found for both the discrete case, in which centers must be on vertices, and the continuous case, in which centers may also be placed on edges. The technique developed also improves previous results for solving the sink evacuation problem on a tree, given the location of the sinks in advance.

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Notes

  1. Confluent flows occur naturally in problems other than evacuations, e.g., packet forwarding and railway scheduling [14].

  2. Because the sinks are predefined, they never move and path monotonicity is superfluous.

  3. It thus needs to check each edge in the tree twice; once in each direction.

  4. (u, v) could have been checked immediately. The deferment is introduced to simplify the later use of parametric searching in Sect. 5.

  5. The version of parametric searching used here is specialized for the case in which the feasibility test \({\mathcal {B}}\) can only determine whether \({\mathcal {T}}^* \le {\mathcal {T}}\) but not whether \({\mathcal {T}}^* ={\mathcal {T}}.\) See [1, p. 415] for more details.

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Acknowledgements

The work of both authors was partially supported by Hong Kong RGC CERG Grant 16208415. The authors would also like to thank Professor Tsunehiko Kameda and the anonymous referees for their comments and suggestions which identified an error in an earlier version of Sect. 3.5, and also helped improve the earlier drafts of this paper.

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    Di Chen & Mordecai Golin

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Chen, D., Golin, M. Minmax Centered k-Partitioning of Trees and Applications to Sink Evacuation with Dynamic Confluent Flows. Algorithmica 85, 1948–2000 (2023). https://doi.org/10.1007/s00453-022-01083-y

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