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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space (V,c) and an activation probability p(v) for each customer v ∈ V. We ask for a TSP tour T for V that minimizes the expected length after cutting T short by skipping the inactive customers.
All known approximation algorithms select a nonempty subset S of the customers and construct a master route solution, consisting of a TSP tour for S and two edges connecting every customer v ∈ V⧵S to a nearest customer in S.
We address the following questions. If we randomly sample the subset S, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a deterministic polynomial-time algorithm.

Jannis Blauth, Meike Neuwohner, Luise Puhlmann, and Jens Vygen. Improved Guarantees for the a Priori TSP. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blauth_et_al:LIPIcs.ISAAC.2023.14, author = {Blauth, Jannis and Neuwohner, Meike and Puhlmann, Luise and Vygen, Jens}, title = {{Improved Guarantees for the a Priori TSP}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {14:1--14:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.14}, URN = {urn:nbn:de:0030-drops-193161}, doi = {10.4230/LIPIcs.ISAAC.2023.14}, annote = {Keywords: A priori TSP, random sampling, stochastic combinatorial optimization} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We introduce the problem of finding a set B of k points in [0,1]ⁿ such that the expected cost of the cheapest point in B that dominates a random point from [0,1]ⁿ is minimized. We study the case where the coordinates of the random points are independently distributed and the cost function is linear. This problem arises naturally in various application areas where customers' requests are satisfied based on predefined products, each corresponding to a subset of features. We show that the problem is NP-hard already for k = 2 when each coordinate is drawn from {0,1}, and obtain an FPTAS for general fixed k under mild assumptions on the distributions.

Bento Natura, Meike Neuwohner, and Stefan Weltge. The Pareto Cover Problem. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 80:1-80:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{natura_et_al:LIPIcs.ESA.2022.80, author = {Natura, Bento and Neuwohner, Meike and Weltge, Stefan}, title = {{The Pareto Cover Problem}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {80:1--80:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.80}, URN = {urn:nbn:de:0030-drops-170186}, doi = {10.4230/LIPIcs.ESA.2022.80}, annote = {Keywords: Pareto, Covering, Optimization, Approximation Algorithm} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

In this paper, we consider the task of computing an independent set of maximum weight in a given d-claw free graph G = (V,E) equipped with a positive weight function w:V → ℝ^+. Thereby, d ≥ 2 is considered a constant. The previously best known approximation algorithm for this problem is the local improvement algorithm SquareImp proposed by Berman [Berman, 2000]. It achieves a performance ratio of d/2+ε in time 𝒪(|V(G)|^(d+1)⋅(|V(G)|+|E(G)|)⋅(d-1)²⋅ (d/(2ε)+1)²) for any ε > 0, which has remained unimproved for the last twenty years. By considering a broader class of local improvements, we obtain an approximation ratio of d/2-(1/63,700,992)+ε for any ε > 0 at the cost of an additional factor of 𝒪(|V(G)|^(d-1)²) in the running time. In particular, our result implies a polynomial time d/2-approximation algorithm. Furthermore, the well-known reduction from the weighted k-Set Packing Problem to the Maximum Weight Independent Set Problem in k+1-claw free graphs provides a (k+1)/2 -(1/63,700,992)+ε-approximation algorithm for the weighted k-Set Packing Problem for any ε > 0. This improves on the previously best known approximation guarantee of (k+1)/2 + ε originating from the result of Berman [Berman, 2000].

Meike Neuwohner. An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 53:1-53:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{neuwohner:LIPIcs.STACS.2021.53, author = {Neuwohner, Meike}, title = {{An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {53:1--53:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.53}, URN = {urn:nbn:de:0030-drops-136982}, doi = {10.4230/LIPIcs.STACS.2021.53}, annote = {Keywords: d-Claw free Graphs, independent Set, local Improvement, k-Set Packing, weighted} }