Minmax-Regret $k$-Sink Location on a Dynamic Tree Network with Uniform Capacities

A dynamic flow network $G$ with uniform capacity $c$ is a graph in which at most $c$ units of flow can enter an edge in one time unit. If flow enters a vertex faster than it can leave, congestion occurs. The evacuation problem is to evacuate all flow to sinks assuming that all flow is confluent, i.e., all flow passing through a particular vertex must follow the same exit edge. The $k$-sink location problem is to place $k$-sinks so as to minimize this evacuation time. Although the $k$-sink location problem is NP-Hard on a general graph it can be solved in $\tilde O(k^2 n)$ time on trees. The concept of minmax-regret arises from robust optimization. For each source, a range of possible flow values is provided and any scenario with flow values in those ranges might occur. The goal is to find a sink placement that minimizes, over all possible scenarios, the difference between the evacuation time to those sinks and the minimal evacuation time of that scenario. The Minmax-Regret $k$-Sink Location on a Dynamic Path Networks with uniform capacities is polynomial solvable in $n$ and $k$. Similarly, the Minmax-Regret $k$-center problem on trees is polynomial solvable in $n$ and $k$. Prior to this work, polynomial time solutions to the Minmax-Regret $k$-Sink Location on Dynamic Tree Networks with uniform capacities were only known for $k=1$. This paper solves this problem, for general $k,$ in time $$O\Bigl( \max(k^2 \log^2 k,\log ^2n)\, k^4 n^2 \log^5 n\Bigr)$$