Facility Location Problem with Capacity Constraints: Algorithmic and Mechanism Design Perspectives
This addresses a practical problem in resource allocation for strategic agents, but it is incremental as it extends known uncapacitated models to include capacity constraints.
The paper tackles the facility location problem with capacity constraints in one dimension, proving NP-hardness for minimizing total or maximum cost but showing fixed-parameter tractability under bounded facilities or identical capacities. It also proposes new strategyproof mechanisms with approximation guarantees nearly matching lower bounds.
We consider the facility location problem in the one-dimensional setting where each facility can serve a limited number of agents from the algorithmic and mechanism design perspectives. From the algorithmic perspective, we prove that the corresponding optimization problem, where the goal is to locate facilities to minimize either the total cost to all agents or the maximum cost of any agent is NP-hard. However, we show that the problem is fixed-parameter tractable, and the optimal solution can be computed in polynomial time whenever the number of facilities is bounded, or when all facilities have identical capacities. We then consider the problem from a mechanism design perspective where the agents are strategic and need not reveal their true locations. We show that several natural mechanisms studied in the uncapacitated setting either lose strategyproofness or a bound on the solution quality for the total or maximum cost objective. We then propose new mechanisms that are strategyproof and achieve approximation guarantees that almost match the lower bounds.