## Abstract

In the Submodular Facility Location problem (SFL) we are given a collection of n clients and m facilities in a metric space. A feasible solution consists of an assignment of each client to some facility. For each client, one has to pay the distance to the associated facility. Furthermore, for each facility f to which we assign the subset of clients S^{f}, one has to pay the opening cost g(S^{f}), where g(·) is a monotone submodular function with g(∅) = 0. SFL is APX-hard since it includes the classical (metric uncapacitated) Facility Location problem (with uniform facility costs) as a special case. Svitkina and Tardos [35, SODA’06] gave the current-best O(log n) approximation algorithm for SFL. The same authors pose the open problem whether SFL admits a constant approximation and provide such an approximation for a very restricted special case of the problem. We make some progress towards the solution of the above open problem by presenting an O(log log n) approximation. Our approach is rather flexible and can be easily extended to generalizations and variants of SFL. In more detail, we achieve the same approximation factor for the natural generalizations of SFL where the opening cost of each facility f is of the form p_{f} + g(S^{f}) or w_{f} · g(S^{f}), where p_{f}, w_{f} ≥ 0 are input values. We also obtain an improved approximation algorithm for the related Universal Stochastic Facility Location problem. In this problem one is given a classical (metric) facility location instance and has to a priori assign each client to some facility. Then a subset of active clients is sampled from some given distribution, and one has to pay (a posteriori) only the connection and opening costs induced by the active clients. The expected opening cost of each facility f can be modelled with a submodular function of the set of clients assigned to f.

Original language | English |
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Title of host publication | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |

Editors | Karl Bringmann, Martin Grohe, Gabriele Puppis, Ola Svensson |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959773225 |

DOIs | |

State | Published - 1 Jul 2024 |

Externally published | Yes |

Event | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 - Tallinn, Estonia Duration: 8 Jul 2024 → 12 Jul 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 297 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |
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Country/Territory | Estonia |

City | Tallinn |

Period | 8/07/24 → 12/07/24 |

## Keywords

- approximation algorithms
- facility location
- submodular facility location
- universal stochastic facility location

## ASJC Scopus subject areas

- Software