An 0.828-approximation algorithm for the uncapacitated facility location problem Original Research Article

The uncapacitated facility location problem in the following formulation is considered:maxS⊆I Z(S)=∑j∈Jmaxi∈Sbij−∑i∈Sci,where I and J are finite sets, and bij, ci⩾0 are rational numbers. Let Z∗ denote the optimal value of the problem and let ZR=∑j∈Jmini∈Ibij−∑i∈Ici. Cornuejols et al. (Ann. Discrete Math. 1 (1977) 163–178) prove that for the problem with the additional cardinality constraint |S|⩽K, a simple greedy algorithm finds a feasible solution S such that (Z(S)−ZR)/(Z∗−ZR)⩾1−e−1≈0.632. We suggest a polynomial-time approximation algorithm for the unconstrained version of the problem, based on the idea of randomized rounding due to Goemans and Williamson (SIAM J. Discrete Math. 7 (1994) 656–666). It is proved that the algorithm delivers a solution S such that (Z(S)−ZR)/(Z∗−ZR)⩾2(2−1)≈0.828. We also show that there exists ε>0 such that it is NP-hard to find an approximate solution S with (Z(S)−ZR)/(Z∗−ZR)⩾1−ε.