Facility Location Problem with Capacity Constraints: Algorithmic and Mechanism Design Perspectives

TL;DR

This paper studies the one-dimensional facility location problem with capacity constraints, proving NP-hardness, fixed-parameter tractability, and designing strategyproof mechanisms with near-optimal guarantees.

Contribution

It introduces new algorithms for fixed-parameter tractability and develops strategyproof mechanisms with strong approximation bounds for capacitated facility location.

Findings

NP-hardness of the capacitated facility location problem

Fixed-parameter tractability when the number of facilities or capacities are bounded

New strategyproof mechanisms with near-optimal approximation guarantees

Abstract

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.