Submodularity Property for Facility Locations of Dynamic Flow Networks

This paper considers facility location problems within dynamic flow networks, shifting the focus from minimizing evacuation time to handling situations with a constrained evacuation timeframe. Our study sets two main goals: 1) Determining a fixed-size set of locations that can maximize the number of evacuees, and 2) Identifying the smallest set of locations capable of accommodating all evacuees within the time constraint. We introduce flow_t(S) to represent the number of evacuees for given locations S within a fixed time limit t. We prove that flow_t functions is a monotone submodular function, which allows us to apply an approximation algorithm specifically designed for maximizing such functions with size restrictions. For the second objective, we implement an approximation algorithm tailored to solving the submodular cover problem. We conduct experiments on the real datasets of Chiang Mai, and demonstrate that the approximation algorithms give solutions which are close to optimal for all of the experiments we have conducted