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

The metric uncapacitated facility location problem (Ufl) enjoys a special stature in approximation algorithms as a testbed for various techniques, among which LP-based methods have been especially prominent and successful. 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. One could hope that different, less natural relaxations might provide better lower bounds. 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 and they are negative in nature. We show unbounded integrality gaps for two substantial families of strengthened formulations. The first family we consider is the hierarchy of LPs resulting from repeated applications of the lift-and-project Lovász-Schrijver procedure starting from the standard relaxation. We show that the LP relaxation for Cfl resulting after Ω(n) rounds, where n is the number of facilities in the instance, has unbounded integrality gap. Note that the Lovász-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 behavior of proper relaxations for both Lbfl and Cfl through a sharp threshold phenomenon under which the integrality gap drops from unbounded to 1. This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) Research Funding Program: “Thalis. Investing in knowledge society through the European Social Fund”. Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Panepistimiopolis Ilissia, Athens 157 84, Greece; (www.di.uoa.gr/ ̃sgk). Part of this work conducted while visiting the IEOR Department, Columbia University, New York, NY 10027. Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Panepistimiopolis Ilissia, Athens 157 84, Greece; ([email protected]). Partially supported by an NKUAELKE graduate fellowship.