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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

We analyze the running time of the Hartigan-Wong method, an old algorithm for the k-means clustering problem. First, we construct an instance on the line on which the method can take 2^{Ω(n)} steps to converge, demonstrating that the Hartigan-Wong method has exponential worst-case running time even when k-means is easy to solve. As this is in contrast to the empirical performance of the algorithm, we also analyze the running time in the framework of smoothed analysis. In particular, given an instance of n points in d dimensions, we prove that the expected number of iterations needed for the Hartigan-Wong method to terminate is bounded by k^{12kd}⋅ poly(n, k, d, 1/σ) when the points in the instance are perturbed by independent d-dimensional Gaussian random variables of mean 0 and standard deviation σ.

Bodo Manthey and Jesse van Rhijn. Worst-Case and Smoothed Analysis of the Hartigan-Wong Method for k-Means Clustering. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{manthey_et_al:LIPIcs.STACS.2024.52, author = {Manthey, Bodo and van Rhijn, Jesse}, title = {{Worst-Case and Smoothed Analysis of the Hartigan-Wong Method for k-Means Clustering}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {52:1--52:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.52}, URN = {urn:nbn:de:0030-drops-197628}, doi = {10.4230/LIPIcs.STACS.2024.52}, annote = {Keywords: k-means clustering, smoothed analysis, probabilistic analysis, local search, heuristics} }

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

The 2-opt heuristic is a simple local search heuristic for the Travelling Salesperson Problem (TSP). Although it usually performs well in practice, its worst-case running time is poor. Attempts to reconcile this difference have used smoothed analysis, in which adversarial instances are perturbed probabilistically. We are interested in the classical model of smoothed analysis for the Euclidean TSP, in which the perturbations are Gaussian. This model was previously used by Manthey & Veenstra, who obtained smoothed complexity bounds polynomial in n, the dimension d, and the perturbation strength σ^{-1}. However, their analysis only works for d ≥ 4. The only previous analysis for d ≤ 3 was performed by Englert, Röglin & Vöcking, who used a different perturbation model which can be translated to Gaussian perturbations. Their model yields bounds polynomial in n and σ^{-d}, and super-exponential in d. As the fact that no direct analysis exists for Gaussian perturbations that yields polynomial bounds for all d is somewhat unsatisfactory, we perform this missing analysis. Along the way, we improve all existing smoothed complexity bounds for Euclidean 2-opt with Gaussian perturbations.

Bodo Manthey and Jesse van Rhijn. Improved Smoothed Analysis of 2-Opt for the Euclidean TSP. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{manthey_et_al:LIPIcs.ISAAC.2023.52, author = {Manthey, Bodo and van Rhijn, Jesse}, title = {{Improved Smoothed Analysis of 2-Opt for the Euclidean TSP}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {52:1--52:16}, 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.2023.52}, URN = {urn:nbn:de:0030-drops-193549}, doi = {10.4230/LIPIcs.ISAAC.2023.52}, annote = {Keywords: Travelling salesman problem, smoothed analysis, probabilistic analysis, local search, heuristics, 2-opt} }

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**Published in:** LIPIcs, Volume 159, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)

Simple heuristics for (combinatorial) optimization problems often show a remarkable performance in practice. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms.
The instances of many (combinatorial) optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained in recent years, where random shortest path metrics generated from dense graphs (either complete graphs or Erdős - Rényi random graphs) have been used so far.
In this paper we extend these findings to sparse graphs, with a focus on grid graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances generated from a grid graph, we prove that the greedy heuristic for the minimum distance maximum matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, for instances generated from an arbitrary sparse graph, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio.

Stefan Klootwijk and Bodo Manthey. Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{klootwijk_et_al:LIPIcs.AofA.2020.19, author = {Klootwijk, Stefan and Manthey, Bodo}, title = {{Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics}}, booktitle = {31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-147-4}, ISSN = {1868-8969}, year = {2020}, volume = {159}, editor = {Drmota, Michael and Heuberger, Clemens}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2020.19}, URN = {urn:nbn:de:0030-drops-120494}, doi = {10.4230/LIPIcs.AofA.2020.19}, annote = {Keywords: Random shortest paths, Random metrics, Approximation algorithms, First-passage percolation} }

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**Published in:** Dagstuhl Reports, Volume 7, Issue 4 (2018)

This report documents the program and the outcomes of Dagstuhl Seminar 17141 "Probabilistic Methods in the Design and Analysis of Algorithms".
Probabilistic methods play a central role in theoretical computer science. They are a powerful and widely applied tool used, for example, for designing efficient randomized algorithms and for establishing various lower bounds in complexity theory. They also form the basis of frameworks like average-case and smoothed analysis, in which algorithms are analyzed beyond the classical worst-case perspective. The seminar was on probabilistic methods with a focus on the design and analysis of algorithms.
The seminar helped to consolidate the research and to foster collaborations among the researchers who use probabilistic methods in different areas of the design and analysis of algorithms.

Bodo Manthey, Claire Mathieu, Heiko Röglin, and Eli Upfal. Probabilistic Methods in the Design and Analysis of Algorithms (Dagstuhl Seminar 17141). In Dagstuhl Reports, Volume 7, Issue 4, pp. 1-22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{manthey_et_al:DagRep.7.4.1, author = {Manthey, Bodo and Mathieu, Claire and R\"{o}glin, Heiko and Upfal, Eli}, title = {{Probabilistic Methods in the Design and Analysis of Algorithms (Dagstuhl Seminar 17141)}}, pages = {1--22}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {7}, number = {4}, editor = {Manthey, Bodo and Mathieu, Claire and R\"{o}glin, Heiko and Upfal, Eli}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.4.1}, URN = {urn:nbn:de:0030-drops-75452}, doi = {10.4230/DagRep.7.4.1}, annote = {Keywords: analysis of algorithms, average-case analysis, random graphs, randomized algorithms, smoothed analysis, sub-linear algorithms} }

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**Published in:** Dagstuhl Reports, Volume 4, Issue 9 (2015)

This report documents the program and the outcomes of Dagstuhl Seminar 14372 "Analysis of Algorithms Beyond the Worst Case".
The theory of algorithms has traditionally focused on worst-case analysis. This focus has led to both a deep theory and many beautiful and useful algorithms. However, there are a number of important problems and algorithms for which worst-case analysis does not provide useful or empirically accurate results. This is due to the fact that worst-case inputs are often rather contrived and occur hardly ever in practical applications. Only in recent years a paradigm shift towards a more realistic and robust algorithmic theory has been initiated. The development of a more realistic theory hinges on finding models that measure the performance of an algorithm not only by its worst-case behavior but rather by its behavior on "typical" inputs. In this seminar, we discussed various recent theoretical models and results that go beyond worst-case analysis.
The seminar helped to consolidate the research and to foster collaborations among the researchers working in the different branches of analysis of algorithms beyond the worst case.

Marina-Florina Balcan, Bodo Manthey, Heiko Röglin, and Tim Roughgarden. Analysis of Algorithms Beyond the Worst Case (Dagstuhl Seminar 14372). In Dagstuhl Reports, Volume 4, Issue 9, pp. 30-49, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Article{balcan_et_al:DagRep.4.9.30, author = {Balcan, Marina-Florina and Manthey, Bodo and R\"{o}glin, Heiko and Roughgarden, Tim}, title = {{Analysis of Algorithms Beyond the Worst Case (Dagstuhl Seminar 14372)}}, pages = {30--49}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2015}, volume = {4}, number = {9}, editor = {Balcan, Marina-Florina and Manthey, Bodo and R\"{o}glin, Heiko and Roughgarden, Tim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.9.30}, URN = {urn:nbn:de:0030-drops-48829}, doi = {10.4230/DagRep.4.9.30}, annote = {Keywords: analysis of algorithms, probabilistic analysis, smoothed analysis, approximation stability, machine learning} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We present approximation algorithms for almost all variants of the multi-criteria traveling salesman problem (TSP), whose performances are independent of the number $k$ of criteria and come close to the approximation ratios obtained for TSP with a single objective function.
We present randomized approximation algorithms for multi-criteria maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP, where the edge weights have to be symmetric, we devise an algorithm that achieves an approximation ratio of $2/3 - \varepsilon$. For multi-criteria Max-ATSP, where the edge weights may be asymmetric, we present an algorithm with an approximation ratio of $1/2 - \varepsilon$. Our algorithms work for any fixed number $k$ of objectives. To get these ratios, we introduce a decomposition technique for cycle covers. These decompositions are optimal in the sense that no decomposition can always yield more than a fraction of $2/3$ and $1/2$, respectively, of the weight of a cycle cover. Furthermore, we present a deterministic algorithm for bi-criteria Max-STSP\ that achieves an approximation ratio of $61/243 \approx 1/4$.
Finally, we present a randomized approximation algorithm for the asymmetric multi-criteria minimum TSP with triangle inequality (Min-ATSP). This algorithm achieves a ratio of $\log n + \varepsilon$. For this variant of multi-criteria TSP, this is the first approximation algorithm we are aware of. If the distances fulfil the $\gamma$-triangle inequality, its ratio is $1/(1-\gamma) + \varepsilon$.

Bodo Manthey. On Approximating Multi-Criteria TSP. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 637-648, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{manthey:LIPIcs.STACS.2009.1853, author = {Manthey, Bodo}, title = {{On Approximating Multi-Criteria TSP}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {637--648}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1853}, URN = {urn:nbn:de:0030-drops-18537}, doi = {10.4230/LIPIcs.STACS.2009.1853}, annote = {Keywords: Approximation algorithms, Traveling salesman, Multi-criteria optimization} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 7391, Probabilistic Methods in the Design and Analysis of Algorithms (2007)

While the height of binary search trees is linear in the worst case, their
average height is logarithmic. We investigate what happens in between, i.e.,
when the randomness is limited, by analyzing the smoothed height of binary
search trees: Randomly perturb a given (adversarial) sequence and then take
the expected height of the binary search tree generated by the resulting
sequence.
As perturbation models, we consider partial permutations, where some
elements are randomly permuted, and additive noise, where random numbers
are added to the adversarial sequence. We prove tight bounds for the
smoothed height of binary search trees under these models. We also obtain
tight bounds for smoothed number of left-to-right maxima. Furthermore, we
exploit the results obtained to get bounds for the smoothed number of
comparisons that quicksort needs.

Bodo Manthey and Till Tantau. Smoothed Analysis of Binary Search Trees and Quicksort Under Additive Noise. In Probabilistic Methods in the Design and Analysis of Algorithms. Dagstuhl Seminar Proceedings, Volume 7391, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{manthey_et_al:DagSemProc.07391.3, author = {Manthey, Bodo and Tantau, Till}, title = {{Smoothed Analysis of Binary Search Trees and Quicksort Under Additive Noise}}, booktitle = {Probabilistic Methods in the Design and Analysis of Algorithms}, pages = {1--19}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {7391}, editor = {Martin Dietzfelbinger and Shang-Hua Teng and Eli Upfal and Berthold V\"{o}cking}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07391.3}, URN = {urn:nbn:de:0030-drops-12893}, doi = {10.4230/DagSemProc.07391.3}, annote = {Keywords: Smoothed Analysis, Binary Search Trees, Quicksort, Left-to-right Maxima} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

Given a set of monomials, the {sc Minimum AND-Circuit} problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size.
We prove that the problem is not polynomial time approximable within a factor of less than $1.0051$ unless $mathsf{P} = mathsf{NP}$, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of $1.278$. For the general problem, we achieve an approximation ratio of $d-3/2$, where $d$ is the degree of the largest monomial.
In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the {sc Minimum AND-Circuit} problem and several problems from different areas.

Jan Arpe and Bodo Manthey. Approximability of Minimum AND-Circuits. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{arpe_et_al:DagSemProc.06111.4, author = {Arpe, Jan and Manthey, Bodo}, title = {{Approximability of Minimum AND-Circuits}}, booktitle = {Complexity of Boolean Functions}, pages = {1--21}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.4}, URN = {urn:nbn:de:0030-drops-6039}, doi = {10.4230/DagSemProc.06111.4}, annote = {Keywords: Optimization problems, approximability, automated circuit design} }

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