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**Published in:** LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Since its introduction in 1957, Lloyd’s algorithm for k-means clustering has been extensively studied and has undergone several improvements. While in its original form it does not guarantee any approximation factor at all, Arthur and Vassilvitskii (SODA 2007) proposed k-means++ which enhances Lloyd’s algorithm by a seeding method which guarantees a 𝒪(log k)-approximation in expectation. More recently, Lattanzi and Sohler (ICML 2019) proposed LS++ which further improves the solution quality of k-means++ by local search techniques to obtain a 𝒪(1)-approximation. On the practical side, the greedy variant of k-means++ is often used although its worst-case behaviour is provably worse than for the standard k-means++ variant.
We investigate how to improve LS++ further in practice. We study two options for improving the practical performance: (a) Combining LS++ with greedy k-means++ instead of k-means++, and (b) Improving LS++ by better entangling it with Lloyd’s algorithm. Option (a) worsens the theoretical guarantees of k-means++ but improves the practical quality also in combination with LS++ as we confirm in our experiments. Option (b) is our new algorithm, Foresight LS++. We experimentally show that FLS++ improves upon the solution quality of LS++. It retains its asymptotic runtime and its worst-case approximation bounds.

Theo Conrads, Lukas Drexler, Joshua Könen, Daniel R. Schmidt, and Melanie Schmidt. Local Search k-means++ with Foresight. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{conrads_et_al:LIPIcs.SEA.2024.7, author = {Conrads, Theo and Drexler, Lukas and K\"{o}nen, Joshua and Schmidt, Daniel R. and Schmidt, Melanie}, title = {{Local Search k-means++ with Foresight}}, booktitle = {22nd International Symposium on Experimental Algorithms (SEA 2024)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-325-6}, ISSN = {1868-8969}, year = {2024}, volume = {301}, editor = {Liberti, Leo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.7}, URN = {urn:nbn:de:0030-drops-203727}, doi = {10.4230/LIPIcs.SEA.2024.7}, annote = {Keywords: k-means clustering, kmeans++, greedy, local search} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graph G on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints.
Our main result is an O(1)-approximation algorithm for the disjoint connected k-center and k-diameter problem for Euclidean spaces of low dimension (constant d) and for metrics with constant doubling dimension. For general metrics, we get an O(log²k)-approximation. Our algorithms work by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering.
We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.

Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Connected k-Center and k-Diameter Clustering. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{drexler_et_al:LIPIcs.ICALP.2023.50, author = {Drexler, Lukas and Eube, Jan and Luo, Kelin and R\"{o}glin, Heiko and Schmidt, Melanie and Wargalla, Julian}, title = {{Connected k-Center and k-Diameter Clustering}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {50:1--50:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.50}, URN = {urn:nbn:de:0030-drops-181024}, doi = {10.4230/LIPIcs.ICALP.2023.50}, annote = {Keywords: Approximation algorithms, Clustering, Connectivity constraints} }

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APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

In a hierarchical clustering problem the task is to compute a series of mutually compatible clusterings of a finite metric space (P,dist). Starting with the clustering where every point forms its own cluster, one iteratively merges two clusters until only one cluster remains. Complete linkage is a well-known and popular algorithm to compute such clusterings: in every step it merges the two clusters whose union has the smallest radius (or diameter) among all currently possible merges. We prove that the radius (or diameter) of every k-clustering computed by complete linkage is at most by factor O(k) (or O(k²)) worse than an optimal k-clustering minimizing the radius (or diameter). Furthermore we give a negative answer to the question proposed by Dasgupta and Long [Sanjoy Dasgupta and Philip M. Long, 2005], who show a lower bound of Ω(log(k)) and ask if the approximation guarantee is in fact Θ(log(k)). We present instances where complete linkage performs poorly in the sense that the k-clustering computed by complete linkage is off by a factor of Ω(k) from an optimal solution for radius and diameter. We conclude that in general metric spaces complete linkage does not perform asymptotically better than single linkage, merging the two clusters with smallest inter-cluster distance, for which we prove an approximation guarantee of O(k).

Anna Arutyunova, Anna Großwendt, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Upper and Lower Bounds for Complete Linkage in General Metric Spaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arutyunova_et_al:LIPIcs.APPROX/RANDOM.2021.18, author = {Arutyunova, Anna and Gro{\ss}wendt, Anna and R\"{o}glin, Heiko and Schmidt, Melanie and Wargalla, Julian}, title = {{Upper and Lower Bounds for Complete Linkage in General Metric Spaces}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {18:1--18:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.18}, URN = {urn:nbn:de:0030-drops-147115}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.18}, annote = {Keywords: Hierarchical Clustering, Complete Linkage, agglomerative Clustering, k-Center} }

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

We study k-clustering problems with lower bounds, including k-median and k-means clustering with lower bounds. In addition to the point set P and the number of centers k, a k-clustering problem with (uniform) lower bounds gets a number B. The solution space is restricted to clusterings where every cluster has at least B points. We demonstrate how to approximate k-median with lower bounds via a reduction to facility location with lower bounds, for which O(1)-approximation algorithms are known.
Then we propose a new constrained clustering problem with lower bounds where we allow points to be assigned multiple times (to different centers). This means that for every point, the clustering specifies a set of centers to which it is assigned. We call this clustering with weak lower bounds. We give an 8-approximation for k-median clustering with weak lower bounds and an O(1)-approximation for k-means with weak lower bounds.
We conclude by showing that at a constant increase in the approximation factor, we can restrict the number of assignments of every point to 2 (or, if we allow fractional assignments, to 1+ε). This also leads to the first bicritera approximation algorithm for k-means with (standard) lower bounds where bicriteria is interpreted in the sense that the lower bounds are violated by a constant factor.
All algorithms in this paper run in time that is polynomial in n and k (and d for the Euclidean variants considered).

Anna Arutyunova and Melanie Schmidt. Achieving Anonymity via Weak Lower Bound Constraints for k-Median and k-Means. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arutyunova_et_al:LIPIcs.STACS.2021.7, author = {Arutyunova, Anna and Schmidt, Melanie}, title = {{Achieving Anonymity via Weak Lower Bound Constraints for k-Median and k-Means}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {7:1--7:17}, 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.2021.7}, URN = {urn:nbn:de:0030-drops-136529}, doi = {10.4230/LIPIcs.STACS.2021.7}, annote = {Keywords: Clustering with Constraints, lower Bounds, k-Means, Anonymity} }

Document

**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

The k-means++ algorithm due to Arthur and Vassilvitskii [David Arthur and Sergei Vassilvitskii, 2007] has become the most popular seeding method for Lloyd’s algorithm. It samples the first center uniformly at random from the data set and the other k-1 centers iteratively according to D²-sampling, i.e., the probability that a data point becomes the next center is proportional to its squared distance to the closest center chosen so far. k-means++ is known to achieve an approximation factor of 𝒪(log k) in expectation.
Already in the original paper on k-means++, Arthur and Vassilvitskii suggested a variation called greedy k-means++ algorithm in which in each iteration multiple possible centers are sampled according to D²-sampling and only the one that decreases the objective the most is chosen as a center for that iteration. It is stated as an open question whether this also leads to an 𝒪(log k)-approximation (or even better). We show that this is not the case by presenting a family of instances on which greedy k-means++ yields only an Ω(𝓁⋅log k)-approximation in expectation where 𝓁 is the number of possible centers that are sampled in each iteration.
Inspired by the negative results, we study a variation of greedy k-means++ which we call noisy k-means++ algorithm. In this variation only one center is sampled in every iteration but not exactly by D²-sampling. Instead in each iteration an adversary is allowed to change the probabilities arising from D²-sampling individually for each point by a factor between 1-ε₁ and 1+ε₂ for parameters ε₁ ∈ [0,1) and ε₂ ≥ 0. We prove that noisy k-means++ computes an 𝒪(log² k)-approximation in expectation. We use the analysis of noisy k-means++ to design a moderately greedy k-means++ algorithm.

Anup Bhattacharya, Jan Eube, Heiko Röglin, and Melanie Schmidt. Noisy, Greedy and Not so Greedy k-Means++. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhattacharya_et_al:LIPIcs.ESA.2020.18, author = {Bhattacharya, Anup and Eube, Jan and R\"{o}glin, Heiko and Schmidt, Melanie}, title = {{Noisy, Greedy and Not so Greedy k-Means++}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {18:1--18:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.18}, URN = {urn:nbn:de:0030-drops-128848}, doi = {10.4230/LIPIcs.ESA.2020.18}, annote = {Keywords: k-means++, greedy, adaptive sampling} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Clustering is a fundamental tool in data mining and machine learning. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters - especially if the data is already biased.
At NIPS 2017, Chierichetti et al. [Flavio Chierichetti et al., 2017] proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair k-center problem and a O(t)-approximation for the fair k-median problem, where t is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair k-center [Clemens Rösner and Melanie Schmidt, 2018].
We extend and improve the known results. Firstly, we give a 5-approximation for the fair k-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives k-center, k-supplier, k-median, k-means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed.

Ioana O. Bercea, Martin Groß, Samir Khuller, Aounon Kumar, Clemens Rösner, Daniel R. Schmidt, and Melanie Schmidt. On the Cost of Essentially Fair Clusterings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bercea_et_al:LIPIcs.APPROX-RANDOM.2019.18, author = {Bercea, Ioana O. and Gro{\ss}, Martin and Khuller, Samir and Kumar, Aounon and R\"{o}sner, Clemens and Schmidt, Daniel R. and Schmidt, Melanie}, title = {{On the Cost of Essentially Fair Clusterings}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {18:1--18:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.18}, URN = {urn:nbn:de:0030-drops-112337}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.18}, annote = {Keywords: approximation, clustering, fairness, LP rounding} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The k-center problem is a classical combinatorial optimization problem which asks to find k centers such that the maximum distance of any input point in a set P to its assigned center is minimized. The problem allows for elegant 2-approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least l clients will be assigned to it. We show how to combine privacy with several other constraints.

Clemens Rösner and Melanie Schmidt. Privacy Preserving Clustering with Constraints. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 96:1-96:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{rosner_et_al:LIPIcs.ICALP.2018.96, author = {R\"{o}sner, Clemens and Schmidt, Melanie}, title = {{Privacy Preserving Clustering with Constraints}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {96:1--96:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.96}, URN = {urn:nbn:de:0030-drops-91008}, doi = {10.4230/LIPIcs.ICALP.2018.96}, annote = {Keywords: Clustering, k-center, Constraints, Privacy, Lower Bounds, Fairness} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

In the Steiner Forest problem, we are given a graph and a collection of source-sink pairs, and the goal is to find a subgraph of minimum total length such that all pairs are connected. The problem is APX-Hard and can be 2-approximated by, e.g., the elegant primal-dual algorithm of Agrawal, Klein, and Ravi from 1995.
We give a local-search-based constant-factor approximation for the problem. Local search brings in new techniques to an area that has for long not seen any improvements and might be a step towards a combinatorial algorithm for the more general survivable network design problem. Moreover, local search was an essential tool to tackle the dynamic MST/Steiner Tree problem, whereas dynamic Steiner Forest is still wide open.
It is easy to see that any constant factor local search algorithm requires steps that add/drop many edges together. We propose natural local moves which, at each step, either (a) add a shortest path in the current graph and then drop a bunch of inessential edges, or (b) add a set of edges to the current solution. This second type of moves is motivated by the potential function we use to measure progress, combining the cost of the solution with a penalty for each connected component. Our carefully-chosen local moves and potential function work in tandem to eliminate bad local minima that arise when using more traditional local moves.
Our analysis first considers the case where the local optimum is a single tree, and shows optimality w.r.t. moves that add a single edge (and drop a set of edges) is enough to bound the locality gap. For the general case, we show how to "project" the optimal solution onto the different trees of the local optimum without incurring too much cost (and this argument uses optimality w.r.t. both kinds of moves), followed by a tree-by-tree argument. We hope both the potential function, and our analysis techniques will be useful to develop and analyze local-search algorithms in other contexts.

Martin Groß, Anupam Gupta, Amit Kumar, Jannik Matuschke, Daniel R. Schmidt, Melanie Schmidt, and José Verschae. A Local-Search Algorithm for Steiner Forest. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gro_et_al:LIPIcs.ITCS.2018.31, author = {Gro{\ss}, Martin and Gupta, Anupam and Kumar, Amit and Matuschke, Jannik and Schmidt, Daniel R. and Schmidt, Melanie and Verschae, Jos\'{e}}, title = {{A Local-Search Algorithm for Steiner Forest}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {31:1--31:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.31}, URN = {urn:nbn:de:0030-drops-83134}, doi = {10.4230/LIPIcs.ITCS.2018.31}, annote = {Keywords: Local Search, Steiner Forest, Approximation Algorithms, Network Design} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

In the aversion k-clustering problem, given a metric space, we want to cluster the points into k clusters. The cost incurred by each point is the distance to the furthest point in its cluster, and the cost of the clustering is the sum of all these per-point-costs. This problem is motivated by questions in generating automatic abstractions of extensive-form games.
We reduce this problem to a "local" k-median problem where each facility has a prescribed radius and can only connect to clients within that radius. Our main results is a constant-factor approximation algorithm for the aversion k-clustering problem via the local k-median problem.
We use a primal-dual approach; our technical contribution is a non-local rounding step which we feel is of broader interest.

Anupam Gupta, Guru Guruganesh, and Melanie Schmidt. Approximation Algorithms for Aversion k-Clustering via Local k-Median. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 66:1-66:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gupta_et_al:LIPIcs.ICALP.2016.66, author = {Gupta, Anupam and Guruganesh, Guru and Schmidt, Melanie}, title = {{Approximation Algorithms for Aversion k-Clustering via Local k-Median}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {66:1--66:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.66}, URN = {urn:nbn:de:0030-drops-62180}, doi = {10.4230/LIPIcs.ICALP.2016.66}, annote = {Keywords: Approximation algorithms, clustering, k-median, primal-dual} }