Multiple sink location problems in dynamic path networks

This paper considers the k-sink location problem in dynamic path networks. A dynamic path network consists of an undirected path with positive edge lengths, uniform edge capacity, and positive vertex supplies. A path can be considered as a road, edge lengths as the distance along a road segment and vertex supplies as the number of people at a location. The edges all have a fixed common capacity, which limits the number of people that can enter that edge in a unit of time. The problem is to find the optimal location of k sinks (exits) on the path such that each evacuee is sent to one of the k sinks. The existence of capacities causes congestion, which can slow evacuation down in unexpected ways.Let x be a vector denoting the location of the k sinks. The optimal evacuation policy for x   is (k−1)(k−1)-dimensional vector d  , called (k−1)(k−1)-divider. Each component of d corresponds to a boundary dividing all evacuees between adjacent two sinks into two groups, i.e., all supplies to the right of the boundary evacuate to the right sink and all the others to the left sink. In this paper, we consider optimality   defined by two different criteria, the minimax criterion and the minisum one. We prove that the minimax problem can be solved in O(kn)O(kn) time and the minisum problem in O(n2⋅min⁡{klog⁡n+log⁡n,2log⁡klog⁡log⁡n}) time, where n is the number of vertices in the given network.