Fairness and Efficiency in Facility Location Problems with Continuous Demands

In the facility location problem with continuous demands, where customers are continuously distributed on an area, a planner wants to locate the facilities and allocate the customers to their closest facilities under the proximity rule. In this work, we focus on the fairness and system efficiency from the facility's perspective. Each facility is assumed to have a preference (represented as valuation function) over the subsets of customers. For the fairness of facilities, we provide approximation guarantees for the proportionality and envy-freeness. For the efficiency, we study the utilitarian and egalitarian social welfare. In addition, we are interested in quantifying the possible trade-offs between meeting the fairness criteria and maximizing social welfare, measured by the price of fairness.