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

TL;DR

This paper introduces a polynomial-time algorithm for the minmax centered k-partitioning problem on trees, with applications to sink evacuation, improving efficiency and generality over previous methods.

Contribution

It presents the first polynomial-time solution for sink evacuation with dynamic flows on trees, applicable to both discrete and continuous center placements.

Findings

First polynomial-time algorithm for sink evacuation problem.

Applicable to both discrete and continuous center placements.

Improves previous results for fixed sink locations.

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 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 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 $O\Bigl(\max(k,\log n) k n \log^4 n\Bigr)$ 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 finding a minmax cost $k$-partition of a tree given the location of the sinks in advance.