First Approximation for Uniform Lower and Upper Bounded Facility Location Problem avoiding violation in Lower Bounds

With growing emphasis on e-commerce marketplace platforms where we have a central platform mediating between the seller and the buyer, it becomes important to keep a check on the availability and profitability of the central store. A store serving too less clients can be non-profitable and a store getting too many orders can lead to bad service to the customers which can be detrimental for the business. In this paper, we study the facility location problem(FL) with upper and lower bounds on the number of clients an open facility serves. Constant factor approximations are known for the restricted variants of the problem with only the upper bounds or only the lower bounds. The only work that deals with bounds on both the sides violates both the bounds \cite{friggstad_et_al_LBUFL}. In this paper, we present the first (constant factor) approximation for the problem violating the upper bound by a factor of %\textcolor{magenta}{$(2 + \epsilon)$} $(5/2)$ without violating the lower bounds when both the lower and the upper bounds are uniform. We first give a tri-criteria (constant factor) approximation violating both the upper and the lower bounds and then get rid of violation in lower bounds by transforming the problem instance to an instance of capacitated facility location problem.