Integrality gaps for strengthened LP relaxations of Capacitated and Lower-Bounded Facility Location

TL;DR

This paper investigates the limitations of LP relaxations for capacitated and lower-bounded facility location problems, demonstrating unbounded integrality gaps for strengthened formulations and providing insights into the potential for exact relaxations.

Contribution

It presents the first evidence of unbounded integrality gaps for strengthened LP relaxations of CFL, including hierarchy-based and proper relaxations, advancing understanding of their limitations.

Findings

Unbounded integrality gap after (n) rounds of Love1sz-Schrijver hierarchy.

Proper relaxations exhibit a threshold phenomenon reducing the gap to 1.

Standard LP relaxations have unbounded gaps for CFL and LBFL.

Abstract

The metric uncapacitated facility location problem (UFL) enjoys a special stature in approximation algorithms as a testbed for various techniques. Two generalizations of UFL are capacitated facility location (CFL) and lower-bounded facility location (LBFL). In the former, every facility has a capacity which is the maximum demand that can be assigned to it, while in the latter, every open facility is required to serve a given minimum amount of demand. Both CFL and LBFL are approximable within a constant factor but their respective natural LP relaxations have an unbounded integrality gap. According to Shmoys and Williamson, the existence of a relaxation-based algorithm for CFL is one of the top 10 open problems in approximation algorithms. In this paper we give the first results on this problem. We provide substantial evidence against the existence of a good LP relaxation for CFL by showing unbounded integrality gaps for two families of strengthened formulations. The first family we consider is the hierarchy of LPs resulting from repeated applications of the lift-and-project Lov\'{a}sz-Schrijver procedure starting from the standard relaxation. We show that the LP relaxation for CFL resulting after $\Omega(n)$ rounds, where $n$ is the number of facilities in the instance, has unbounded integrality gap. Note that the Lov\'{a}sz-Schrijver procedure is known to yield an exact formulation for CFL in at most $n$ rounds. We also introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation, an equivalent form of the natural LP. We characterize the integrality gap of proper relaxations for both LBFL and CFL and show a threshold phenomenon under which it decreases from unbounded to 1.