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An O(loglog n)-Approximation for Submodular Facility Location

Authors Fateme Abbasi , Marek Adamczyk , Miguel Bosch-Calvo , Jarosław Byrka , Fabrizio Grandoni , Krzysztof Sornat , Antoine Tinguely

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Author Details

Fateme Abbasi
  • University of Wrocław, Poland
Marek Adamczyk
  • University of Wrocław, Poland
Miguel Bosch-Calvo
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Jarosław Byrka
  • University of Wrocław, Poland
Fabrizio Grandoni
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Krzysztof Sornat
  • AGH University, Kraków, Poland
Antoine Tinguely
  • IDSIA, USI-SUPSI, Lugano, Switzerland

Funding

Fateme Abbasi and Jarosław Byrka were supported by Polish National Science Centre (NCN) Grant 2020/39/B/ST6/01641. Marek Adamczyk was supported by Polish National Science Centre (NCN) Grant 2019/35/D/ST6/03060. Miguel Bosch-Calvo, Fabrizio Grandoni, Krzysztof Sornat and Antoine Tinguely were supported by the SNSF Grant 200021_200731/1. Krzysztof Sornat was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 101002854).

Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments, in particular for pointing out to the simpler and stronger lower bound construction by Gupta [Shalmoli Gupta, 2018]. We would also like to thank Neil Olver for inspiring discussions about applications of their technique in [Thomas Bosman and Neil Olver, 2020] to various covering problems over time.

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Fateme Abbasi, Marek Adamczyk, Miguel Bosch-Calvo, Jarosław Byrka, Fabrizio Grandoni, Krzysztof Sornat, and Antoine Tinguely. An O(loglog n)-Approximation for Submodular Facility Location. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.5

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(emptyset)=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 [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(loglog 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
  • Theory of computation → Rounding techniques
  • Theory of computation → Online algorithms
Keywords
  • approximation algorithms
  • facility location
  • submodular facility location
  • universal stochastic facility location

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