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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We continue the study of δ-dispersion, a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that every two facilities have distance at least δ from each other.
Our main technical contribution is an efficient procedure to "round-up" distance δ. It transforms a δ-dispersed set S into a δ^⋆-dispersed set S^⋆ of same size where distance δ^⋆ is a potentially slightly larger rational a/b with a numerator a upper bounded by the longest (not-induced) path in the input graph.
Based on this rounding procedure and connections to the distance-d independent set problem we derive a number of algorithmic results. When parameterized by treewidth, the problem is in XP. When parameterized by treedepth the problem is FPT and has a matching lower bound on its time complexity under ETH. Moreover, we can also settle the parameterized complexity with the solution size as parameter using our rounding technique: δ-Dispersion is FPT for every δ ≤ 2 and W[1]-hard for every δ > 2.
Further, we show that δ-dispersion is NP-complete for every fixed irrational distance δ, which was left open in a previous work.

Tim A. Hartmann and Stefan Lendl. Dispersing Obnoxious Facilities on Graphs by Rounding Distances. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hartmann_et_al:LIPIcs.MFCS.2022.55, author = {Hartmann, Tim A. and Lendl, Stefan}, title = {{Dispersing Obnoxious Facilities on Graphs by Rounding Distances}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {55:1--55:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.55}, URN = {urn:nbn:de:0030-drops-168536}, doi = {10.4230/LIPIcs.MFCS.2022.55}, annote = {Keywords: facility location, parameterized complexity, packing} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

We investigate the so-called recoverable robust assignment problem on complete bipartite graphs, a mainstream problem in robust optimization: For two given linear cost functions c₁ and c₂ on the edges and a given integer k, the goal is to find two perfect matchings M₁ and M₂ that minimize the objective value c₁(M₁)+c₂(M₂), subject to the constraint that M₁ and M₂ have at least k edges in common.
We derive a variety of results on this problem. First, we show that the problem is W[1]-hard with respect to parameter k, and also with respect to the complementary parameter k' = n/2-k. This hardness result holds even in the highly restricted special case where both cost functions c₁ and c₂ only take the values 0 and 1. (On the other hand, containment of the problem in XP is straightforward to see.) Next, as a positive result we construct a polynomial time algorithm for the special case where one cost function is Monge, whereas the other one is Anti-Monge. Finally, we study the variant where matching M₁ is frozen, and where the optimization goal is to compute the best corresponding matching M₂. This problem variant is known to be contained in the randomized parallel complexity class RNC², and we show that it is at least as hard as the infamous problem Exact Red-Blue Matching in Bipartite Graphs whose computational complexity is a long-standing open problem.

Dennis Fischer, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger. An Investigation of the Recoverable Robust Assignment Problem. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fischer_et_al:LIPIcs.IPEC.2021.19, author = {Fischer, Dennis and Hartmann, Tim A. and Lendl, Stefan and Woeginger, Gerhard J.}, title = {{An Investigation of the Recoverable Robust Assignment Problem}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.19}, URN = {urn:nbn:de:0030-drops-154025}, doi = {10.4230/LIPIcs.IPEC.2021.19}, annote = {Keywords: assignment problem, matchings, exact matching, robust optimization, fixed paramter tractablity, RNC} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance delta from each other.
We investigate the complexity of this problem in terms of the rational parameter delta. The problem is polynomially solvable, if the numerator of delta is 1 or 2, while all other cases turn out to be NP-hard.

Alexander Grigoriev, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger. Dispersing Obnoxious Facilities on a Graph. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 33:1-33:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{grigoriev_et_al:LIPIcs.STACS.2019.33, author = {Grigoriev, Alexander and Hartmann, Tim A. and Lendl, Stefan and Woeginger, Gerhard J.}, title = {{Dispersing Obnoxious Facilities on a Graph}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {33:1--33:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.33}, URN = {urn:nbn:de:0030-drops-102729}, doi = {10.4230/LIPIcs.STACS.2019.33}, annote = {Keywords: algorithms, complexity, optimization, graph theory, facility location} }

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