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 $O(knlog^4n)$ time algorithm when the edge capacities are non-uniform, where $n$ is the number of vertices. We then present an $O(knlog^3 n)$ time algorithm when the edge capacities are uniform. We also present an $O(nlog n)$ time algorithm for the special case where $k=1$ and the edge capacities are non-uniform.
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