### Abstract

We study the k-level uncapacitated facility location problem (k-level UFL) in which clients need to be connected with paths crossing open facilities of k types (levels). In this paper we first propose an approximation algorithm that for any constant k, in polynomial time, delivers solutions of cost at most α_{k} times OPT, where α_{k} is an increasing function of k, with lim_{k→∞}α_{k}=3. Our algorithm rounds a fractional solution to an extended LP formulation of the problem. The rounding builds upon the technique of iteratively rounding fractional solutions on trees (Garg, Konjevod, and Ravi SODA’98) originally used for the group Steiner tree problem. We improve the approximation ratio for k-level UFL for all k ≥ 3, in particular we obtain the ratio equal 2.02, 2.14, and 2.24 for k = 3,4, and 5. Second, we give a simple interpretation of the randomization process (Li ICALP’2011) for 1-level UFL in terms of solving an auxiliary (factor revealing) LP. Armed with this simple view point, we exercise the randomization on our algorithm for the k-level UFL. We further improve the approximation ratio for all k ≥ 3, obtaining 1.97, 2.09, and 2.19 for k = 3,4, and 5. Third, we extend our algorithm to the k-level UFL with penalties (k-level UFLWP), in which the setting is the same as k-level UFL except that the planner has the option to pay a penalty instead of connecting chosen clients.

Original language | English |
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Pages (from-to) | 19-44 |

Number of pages | 26 |

Journal | Theory of Computing Systems |

Volume | 58 |

Issue number | 1 |

DOIs | |

Publication status | Published - 4 Nov 2014 |

### Keywords

- Approximation algorithms
- Facility location

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## Cite this

*k*-level Uncapacitated Facility Location Problem (with Penalties).

*Theory of Computing Systems*,

*58*(1), 19-44. https://doi.org/10.1007/s00224-014-9575-3