8 Search Results for "Makarychev, Yury"

Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.11
Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment

Authors: Eden Chlamtáč, Yury Makarychev, and Ali Vakilian

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
We provide new approximation algorithms for the Red-Blue Set Cover and Circuit Minimum Monotone Satisfying Assignment (MMSA) problems. Our algorithm for Red-Blue Set Cover achieves Õ(m^{1/3})-approximation improving on the Õ(m^{1/2})-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSA_t (for circuits of depth t) gives an Õ(N^{1-δ}) approximation for δ = 1/32^{3-⌈t/2⌉}, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSA_t with t ≥ 4 were previously known. We complement these results with lower bounds for these problems: For Red-Blue Set Cover, we provide a nearly approximation preserving reduction from Min k-Union that gives an ̃Ω(m^{1/4 - ε}) hardness under the Dense-vs-Random conjecture, while for MMSA we sketch a proof that an SDP relaxation strengthened by Sherali-Adams has an integrality gap of N^{1-ε} where ε → 0 as the circuit depth t → ∞.
Cite as

Eden Chlamtáč, Yury Makarychev, and Ali Vakilian. Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chlamtac_et_al:LIPIcs.APPROX/RANDOM.2023.11,
  author =	{Chlamt\'{a}\v{c}, Eden and Makarychev, Yury and Vakilian, Ali},
  title =	{{Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.11},
  URN =		{urn:nbn:de:0030-drops-188366},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.11},
  annote =	{Keywords: Red-Blue Set Cover Problem, Circuit Minimum Monotone Satisfying Assignment (MMSA) Problem, LP Rounding}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.32
Approximation Algorithm for Norm Multiway Cut

Authors: Charlie Carlson, Jafar Jafarov, Konstantin Makarychev, Yury Makarychev, and Liren Shan

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.13
Two-Sided Kirszbraun Theorem

Authors: Arturs Backurs, Sepideh Mahabadi, Konstantin Makarychev, and Yury Makarychev

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

Abstract
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)‖.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.49
Certified Algorithms: Worst-Case Analysis and Beyond

Authors: Konstantin Makarychev and Yury Makarychev

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.7
Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem

Authors: Haris Angelidakis, Pranjal Awasthi, Avrim Blum, Vaggos Chatziafratis, and Chen Dan

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is gamma-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most gamma >= 1. The goal then is to efficiently recover this "pronounced" optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving O~(Delta/sqrt(log Delta))-stable instances on graphs of maximum degree Delta, (k - 1)-stable instances on k-colorable graphs and (1 + epsilon)-stable instances on planar graphs (for any fixed epsilon > 0), using both combinatorial techniques as well as LPs and the Sherali-Adams hierarchy. For general graphs, we present a strong lower bound showing that there are no efficient algorithms for O(n^(1/2 - epsilon))-stable instances of Independent Set, assuming the planted clique conjecture. To complement our negative result, we give an algorithm for (epsilon n)-stable instances, for any fixed epsilon > 0. As a by-product of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a gamma-certified algorithm was introduced very recently by Makarychev and Makarychev (2018) and it is a class of gamma-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of gamma >= 1 (hence, such algorithms not only solve gamma-stable instances optimally, but also have guarantees even on unstable instances). Here, we obtain Delta-certified algorithms for Independent Set on graphs of maximum degree Delta, and (1+epsilon)-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a ((Delta + 1)/3 + epsilon)-certified algorithm for Independent Set on graphs of maximum degree Delta where all weights are equal to 1.
Cite as

Haris Angelidakis, Pranjal Awasthi, Avrim Blum, Vaggos Chatziafratis, and Chen Dan. Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{angelidakis_et_al:LIPIcs.ESA.2019.7,
  author =	{Angelidakis, Haris and Awasthi, Pranjal and Blum, Avrim and Chatziafratis, Vaggos and Dan, Chen},
  title =	{{Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.7},
  URN =		{urn:nbn:de:0030-drops-111288},
  doi =		{10.4230/LIPIcs.ESA.2019.7},
  annote =	{Keywords: Bilu-Linial stability, perturbation resilience, beyond worst-case analysis, Independent Set, Vertex Cover, Multiway Cut}
}
Document
DOI: 10.4230/DFU.Vol7.15301.287
Approximation Algorithms for CSPs

Authors: Konstantin Makarychev and Yury Makarychev

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.14
A Bi-Criteria Approximation Algorithm for k-Means

Authors: Konstantin Makarychev, Yury Makarychev, Maxim Sviridenko, and Justin Ward

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.339
Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion

Authors: Anand Louis and Yury Makarychev

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which, for a parameter 's' such that 0 < s < 1/2, asks to compute the cut having the least expansion while having at most 's' fraction of the vertices on the smaller side of the cut. We present two algorithms. Our first algorithm gives a multiplicative polylogarithmic approximation. Our second algorithm gives a bound that is a function of the expansion of the hypergraph but is independent of the size of the hypergraph. Using these results, we also obtain similar guarantees for the Small Set Vertex Expansion problem.
Cite as

Anand Louis and Yury Makarychev. Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 339-355, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{louis_et_al:LIPIcs.APPROX-RANDOM.2014.339,
  author =	{Louis, Anand and Makarychev, Yury},
  title =	{{Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{339--355},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.339},
  URN =		{urn:nbn:de:0030-drops-47079},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.339},
  annote =	{Keywords: Approximation Algorithms, Graph Expansion, Hypergraph Expansion, Vertex Expansion}
}
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