Tight bounds on the Lovász-Schrijver rank for approximate Capacitated Facility Location∗

In the Capacitated facility location (Cfl) problem we are given a set F of facilities and a set C of clients in a common metric space. Every facility has a hard capacity ui. Opening a facility i incurs a nonnegative cost fi, while assigning a client j to facility i incurs a nonnegative connection cost cij . The goal is to open a subset F ′ ⊆ F of the facilities and assign each client to an open facility so that at most ui clients are assigned to facility i and the total cost is minimized. Cfl is approximable within a constant factor via local search but the natural LP relaxation has an unbounded integrality gap. It is an open question whether there is a relaxation-based approximation algorithm for Cfl or even if an LP with constant integrality gap exists. We present the first study of a comprehensive family of strengthened linear programs for this problem. We consider the hierarchy of polytopes resulting from repeated applications of the Lovász-Schrijver lift-and-project operator on the standard fractional polytope. We show that the LP relaxation for Cfl resulting in Ω(n) rounds, where n is the number of facilities in the instance, has an unbounded integrality gap. The Lovász-Schrijver procedure is known to yield an exact formulation for Cfl in at most n rounds. ∗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.