[Paper Review] Approximation Algorithms for Hard Capacitated $k$-facility Location Problems

Karen Aardal, Pieter van den Berg+2 more|arXiv (Cornell University)|Nov 19, 2013

Facility Location and Emergency Management18 references6 citations

TL;DR

This paper presents the first fully polynomial-time approximation scheme (FPTAS) for the single-sink capacitated $k$-facility location problem and a polynomial-time algorithm for the uniform-capacity case with a fixed number of clients. It also develops improved approximation algorithms for the capacitated facility location problem with uniform opening costs by leveraging pseudo-approximation techniques and primal-dual methods, achieving a $4.562(1+\epsilon)$-approximation ratio.

ABSTRACT

We study the capacitated $k$-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a constant number $k$. It costs $f_i$ to open facility $i$, and $c_{ij}$ for facility $i$ to serve one unit of demand from client $j$. The objective is to open at most $k$ facilities serving all the demands and satisfying the capacity constraints while minimizing the sum of service and opening costs. In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated $k$-facility location problem. Then, we show that the capacitated $k$-facility location problem with uniform capacities is solvable in polynomial time if the number of clients is fixed by reducing it to a collection of transportation problems. Third, we analyze the structure of extreme point solutions, and examine the efficiency of this structure in designing approximation algorithms for capacitated $k$-facility location problems. Finally, we extend our results to obtain an improved approximation algorithm for the capacitated facility location problem with uniform opening cost.

Motivation & Objective

• To design the first fully polynomial-time approximation scheme (FPTAS) for the single-sink capacitated $k$-facility location problem.
• To develop a polynomial-time algorithm for the uniform-capacity $k$-facility location problem when the number of clients is fixed.
• To analyze the structure of extreme point solutions to guide approximation algorithm design for capacitated $k$-facility location problems.
• To extend results to improve the approximation ratio for the capacitated facility location problem with uniform opening costs.

Proposed method

• Proposes an FPTAS for the single-sink capacitated $k$-facility location problem using dynamic programming and rounding techniques.
• Reduces the uniform-capacity $k$-facility location problem with a fixed number of clients to a collection of transportation problems solvable in polynomial time.
• Analyzes extreme point solutions of the LP relaxation to exploit structural properties for approximation algorithm design.
• Combines a $(\beta, \delta)$-approximation algorithm for the $k$-median problem with a pseudo-approximation framework to handle facility opening and capacity constraints.
• Applies a local search heuristic with scaling from Charikar and Guha to achieve a bi-criteria approximation for the $k$-median problem.
• Uses the resulting approximation to derive a $4.562(1+\epsilon)$-approximation algorithm for the capacitated facility location problem with uniform opening costs.

Experimental results

Research questions

• RQ1 Can a fully polynomial-time approximation scheme (FPTAS) be designed for the single-sink capacitated $k$-facility location problem?
• RQ2 Is the uniform-capacity $k$-facility location problem solvable in polynomial time when the number of clients is fixed?
• RQ3 How can the structure of extreme point solutions in the LP relaxation inform the design of approximation algorithms for capacitated $k$-facility location problems?
• RQ4 Can pseudo-approximation algorithms for $k$-facility location be extended to improve approximation ratios for the general capacitated facility location problem?

Key findings

• An FPTAS is developed for the single-sink capacitated $k$-facility location problem, providing arbitrarily close approximation in fully polynomial time.
• The uniform-capacity $k$-facility location problem with a fixed number of clients is solvable in polynomial time by reducing it to a set of transportation problems.
• The structure of extreme point solutions in the LP relaxation is analyzed, revealing properties useful for approximation algorithm design.
• A $4.562(1+\epsilon)$-approximation algorithm is achieved for the capacitated facility location problem with uniform opening costs, improving upon prior results.
• The use of a $(\beta, \delta)$-approximation for the $k$-median problem enables a bi-criteria approximation that allows up to $\delta k$ facilities to be opened, facilitating improved bounds.
• By combining a pseudo-approximation for $k$-median with a primal-dual framework, the paper achieves a constant-factor approximation for the general capacitated facility location problem.