Minsum $k$-Sink Problem on Path Networks

TL;DR

This paper introduces efficient algorithms for the $k$-sink evacuation problem on path networks with various capacity conditions, optimizing evacuation times through novel computational methods.

Contribution

It provides the first polynomial-time algorithms for the $k$-sink problem on paths with both uniform and non-uniform capacities, improving computational efficiency.

Findings

Algorithm for non-uniform capacities runs in $O(kn ext{log}^4 n)$ time.

Algorithm for uniform capacities runs in $O(kn ext{log}^3 n)$ time.

Special case with $k=1$ achieves $O(n ext{log} n)$ time complexity.

Abstract

We consider the problem of locating a set of $k$ sinks on a path network with general edge capacities that minimizes the sum of the evacuation times of all evacuees. We first present an $O(kn\log^4n)$ time algorithm when the edge capacities are non-uniform, where $n$ is the number of vertices. We then present an $O(kn\log^3 n)$ time algorithm when the edge capacities are uniform. We also present an $O(n\log n)$ time algorithm for the special case where $k=1$ and the edge capacities are non-uniform.