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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph G into k parts so as to separate k given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced 𝓁_p-norm Multiway Cut, a generalization of the problem, in which the goal is to minimize the 𝓁_p norm of the edge boundaries of k parts. We provide an O(log^{1/2} nlog^{1/2+1/p} k) approximation algorithm for this problem, improving upon the approximation guarantee of O(log^{3/2} n log^{1/2} k) due to Chandrasekaran and Wang.
We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an O(log^{1/2} n log^{7/2} k) approximation algorithm with a weaker oracle and an O(log^{1/2} n log^{5/2} k) approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no n^{1/4-ε} approximation algorithm for every ε > 0 assuming the Hypergraph Dense-vs-Random Conjecture.

Charlie Carlson, Jafar Jafarov, Konstantin Makarychev, Yury Makarychev, and Liren Shan. Approximation Algorithm for Norm Multiway Cut. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 32:1-32:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{carlson_et_al:LIPIcs.ESA.2023.32, author = {Carlson, Charlie and Jafarov, Jafar and Makarychev, Konstantin and Makarychev, Yury and Shan, Liren}, title = {{Approximation Algorithm for Norm Multiway Cut}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.32}, URN = {urn:nbn:de:0030-drops-186854}, doi = {10.4230/LIPIcs.ESA.2023.32}, annote = {Keywords: Multiway cut, Approximation algorithms} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y.
In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖.

Arturs Backurs, Sepideh Mahabadi, Konstantin Makarychev, and Yury Makarychev. Two-Sided Kirszbraun Theorem. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 13:1-13:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{backurs_et_al:LIPIcs.SoCG.2021.13, author = {Backurs, Arturs and Mahabadi, Sepideh and Makarychev, Konstantin and Makarychev, Yury}, title = {{Two-Sided Kirszbraun Theorem}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.13}, URN = {urn:nbn:de:0030-drops-138129}, doi = {10.4230/LIPIcs.SoCG.2021.13}, annote = {Keywords: Kirszbraun theorem, Lipschitz map, Outer-extension, Two-sided extension} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a γ-certified algorithm is also a γ-approximation algorithm - it finds a γ-approximation no matter what the input is. Second, it exactly solves γ-perturbation-resilient instances (γ-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient instances, and solve optimization problems with hard constraints.
In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results.

Konstantin Makarychev and Yury Makarychev. Certified Algorithms: Worst-Case Analysis and Beyond. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 49:1-49:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{makarychev_et_al:LIPIcs.ITCS.2020.49, author = {Makarychev, Konstantin and Makarychev, Yury}, title = {{Certified Algorithms: Worst-Case Analysis and Beyond}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.49}, URN = {urn:nbn:de:0030-drops-117347}, doi = {10.4230/LIPIcs.ITCS.2020.49}, annote = {Keywords: certified algorithm, perturbation resilience, Bilu, Linial stability, beyond-worst-case analysis, approximation algorithm, integrality} }

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**Published in:** Dagstuhl Follow-Ups, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability (2017)

In this survey, we offer an overview of approximation algorithms for constraint satisfaction problems (CSPs) - we describe main results and discuss various techniques used for solving CSPs.

Konstantin Makarychev and Yury Makarychev. Approximation Algorithms for CSPs. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 287-325, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InCollection{makarychev_et_al:DFU.Vol7.15301.287, author = {Makarychev, Konstantin and Makarychev, Yury}, title = {{Approximation Algorithms for CSPs}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {287--325}, series = {Dagstuhl Follow-Ups}, ISBN = {978-3-95977-003-3}, ISSN = {1868-8977}, year = {2017}, volume = {7}, editor = {Krokhin, Andrei and Zivny, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DFU.Vol7.15301.287}, URN = {urn:nbn:de:0030-drops-69685}, doi = {10.4230/DFU.Vol7.15301.287}, annote = {Keywords: Constraint satisfaction problems, Approximation algorithms, SDP, UGC} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We consider the classical k-means clustering problem in the setting of bi-criteria approximation, in which an algorithm is allowed to output beta*k > k clusters, and must produce a clustering with cost at most alpha times the to the cost of the optimal set of k clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor.
We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee alpha(beta) depending on the number beta*k of clusters that may be opened. Our guarantee alpha(beta) is always at most 9 + epsilon and improves rapidly with beta (for example: alpha(2) < 2.59, and alpha(3) < 1.4). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.

Konstantin Makarychev, Yury Makarychev, Maxim Sviridenko, and Justin Ward. A Bi-Criteria Approximation Algorithm for k-Means. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 14:1-14:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{makarychev_et_al:LIPIcs.APPROX-RANDOM.2016.14, author = {Makarychev, Konstantin and Makarychev, Yury and Sviridenko, Maxim and Ward, Justin}, title = {{A Bi-Criteria Approximation Algorithm for k-Means}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {14:1--14:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.14}, URN = {urn:nbn:de:0030-drops-66370}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.14}, annote = {Keywords: k-means clustering, bicriteria approximation algorithms, linear programming, local search} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We prove that the Bounded Occurrence Ordering k-CSP Problem is not approximation resistant. We give a very simple local search algorithm that always performs better than the random assignment algorithm (unless, the number of satisfied constraints does not depend on the ordering). Specifically, the expected value of the solution returned by the algorithm is at least ALG >= AVG + alpha(B,k)(OPT-AVG), where OPT is the value of the optimal solution; AVG is the expected value of the random solution; and alpha(B,k) = Omega_k(B^{-(k+O(1))}) is a parameter depending only on k (the arity of the CSP) and B (the maximum number of times each variable is used in constraints).
The question whether bounded occurrence ordering k-CSPs are approximation resistant was raised by Guruswami and Zhou (2012), who recently showed that bounded occurrence 3-CSPs and "monotone" k-CSPs admit a non-trivial approximation.

Konstantin Makarychev. Local Search is Better than Random Assignment for Bounded Occurrence Ordering k-CSPs. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 139-147, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{makarychev:LIPIcs.STACS.2013.139, author = {Makarychev, Konstantin}, title = {{Local Search is Better than Random Assignment for Bounded Occurrence Ordering k-CSPs}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {139--147}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.139}, URN = {urn:nbn:de:0030-drops-39290}, doi = {10.4230/LIPIcs.STACS.2013.139}, annote = {Keywords: approximation algorithms, approximation resistance, ordering CSPs} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

This paper studies the ``explanation problem'' for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A=(a_{ij}) whose rows and columns have semantics: special subsets of the rows and special subsets of the columns are meaningful, others are not. A submatrix in A is said to be meaningful if and only if it is the cross product of a meaningful row subset and a meaningful column subset, in which case we call it an ``allowed rectangle.'' The goal is to ``explain'' A as a sparse sum of weighted allowed rectangles. Specifically, we wish to find as few weighted allowed rectangles as possible such that, for all i,j, a_ij equals the sum of the weights of all rectangles which include cell (i,j).
In this paper we consider the natural cases in which the matrix dimensions are tree-ordered or linearly-ordered. In the tree-ordered case, we are given a rooted tree $T_1$ whose leaves are the rows of $A$ and another, $T_2$, whose leaves are the columns. Nodes of the trees correspond in an obvious way to the sets of their leaf descendants. In the linearly-ordered case, a set of rows or columns is meaningful if and only if it is contiguous.
For tree-ordered data, we prove the explanation problem NP-Hard and give a randomized $2$-approximation algorithm for it. For linearly-ordered data, we prove the explanation problem NP-Har and give a $2.56$-approximation algorithm. To our knowledge, these are the first results for the problem of sparsely and exactly representing matrices by weighted rectangles.

Howard Karloff, Flip Korn, Konstantin Makarychev, and Yuval Rabani. On Parsimonious Explanations For 2-D Tree- and Linearly-Ordered Data. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 332-343, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{karloff_et_al:LIPIcs.STACS.2011.332, author = {Karloff, Howard and Korn, Flip and Makarychev, Konstantin and Rabani, Yuval}, title = {{On Parsimonious Explanations For 2-D Tree- and Linearly-Ordered Data}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {332--343}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.332}, URN = {urn:nbn:de:0030-drops-30246}, doi = {10.4230/LIPIcs.STACS.2011.332}, annote = {Keywords: ordered data, explanation problem} }

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