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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 maximumcost of a subtree over all possible partitions. In the centered version of theproblem, the cost of a subtree cost 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 aconfluent dynamic network flow to evacuate all supplies to sinks.  This paper provides the first polynomial-time algorithm for solving thisproblem, running in $O\Bigl(\max(k,\log n) k n \log^4 n\Bigr)$ time. Thetechnique developed can be used to solve any Minmax Centered $k$-Partitioningproblem on trees in which the servicing costs satisfy some very generalconditions. Solutions can be found for both the discrete case, in which centersmust be on vertices, and the continuous case, in which centers may also beplaced on edges. The technique developed also improves previous results forfinding a minmax cost $k$-partition of a tree given the location of the sinksin advance.

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