9 Search Results for "Bartal, Yair"


Document
Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures

Authors: Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)


Abstract
It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in [Yair Bartal et al., 2019] (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has 1+ε average distortion for embedding into k-dimensional Euclidean space, where k = O(1/ε²), and for more general q-norms of distortion, k = O(max{1/ε²,q/ε}), whereas tight lower bounds were established only for large values of q via reduction to the worst case. In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any 1 ≤ q ≤ O((log (2ε² n))/ε) and ε ≥ 1/(√n), where n is the size of the subset of Euclidean space. Our results also imply that the JL method is optimal for various distortion measures commonly used in practice, such as stress, energy and relative error. We prove that if any of these measures is bounded by ε then k = Ω(1/ε²), for any ε ≥ 1/(√n), matching the upper bounds of [Yair Bartal et al., 2019] and extending their tightness results for the full range moment analysis. Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.

Cite as

Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen. Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 13:1-13:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bartal_et_al:LIPIcs.SoCG.2022.13,
  author =	{Bartal, Yair and Fandina, Ora Nova and Larsen, Kasper Green},
  title =	{{Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.13},
  URN =		{urn:nbn:de:0030-drops-160219},
  doi =		{10.4230/LIPIcs.SoCG.2022.13},
  annote =	{Keywords: average distortion, practical dimensionality reduction, JL transform}
}
Document
Track A: Algorithms, Complexity and Games
Covering Metric Spaces by Few Trees

Authors: Yair Bartal, Nova Fandina, and Ofer Neiman

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)


Abstract
A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y in X has a low distortion path in one of the trees. If it has the stronger property that every point x in X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by [S. Arya et al., 1995]. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

Cite as

Yair Bartal, Nova Fandina, and Ofer Neiman. Covering Metric Spaces by Few Trees. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 20:1-20:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bartal_et_al:LIPIcs.ICALP.2019.20,
  author =	{Bartal, Yair and Fandina, Nova and Neiman, Ofer},
  title =	{{Covering Metric Spaces by Few Trees}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{20:1--20:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.20},
  URN =		{urn:nbn:de:0030-drops-105967},
  doi =		{10.4230/LIPIcs.ICALP.2019.20},
  annote =	{Keywords: tree cover, Ramsey tree cover, probabilistic hierarchical family}
}
Document
Dimension Reduction Techniques for l_p (1<p<2), with Applications

Authors: Yair Bartal and Lee-Ad Gottlieb

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)


Abstract
For Euclidean space (l_2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [Conf. in modern analysis and probability, AMS 1984], with a host of known applications. Here, we consider the problem of dimension reduction for all l_p spaces 1<p<2. Although strong lower bounds are known for dimension reduction in l_1, Ostrovsky and Rabani [JACM 2002] successfully circumvented these by presenting an l_1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to l_1 and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1<p<2, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for l_1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.

Cite as

Yair Bartal and Lee-Ad Gottlieb. Dimension Reduction Techniques for l_p (1<p<2), with Applications. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 16:1-16:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{bartal_et_al:LIPIcs.SoCG.2016.16,
  author =	{Bartal, Yair and Gottlieb, Lee-Ad},
  title =	{{Dimension Reduction Techniques for l\underlinep (1\langlep\langle2), with Applications}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.16},
  URN =		{urn:nbn:de:0030-drops-59081},
  doi =		{10.4230/LIPIcs.SoCG.2016.16},
  annote =	{Keywords: Dimension reduction, embeddings}
}
Document
Minimizing Weighted lp-Norm of Flow-Time in the Rejection Model

Authors: Anamitra Roy Choudhury, Syamantak Das, and Amit Kumar

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)


Abstract
We consider the online scheduling problem to minimize the weighted ell_p-norm of flow-time of jobs. We study this problem under the rejection model introduced by Choudhury et al. (SODA 2015) - here the online algorithm is allowed to not serve an eps-fraction of the requests. We consider the restricted assignments setting where each job can go to a specified subset of machines. Our main result is an immediate dispatch non-migratory 1/eps^{O(1)}-competitive algorithm for this problem when one is allowed to reject at most eps-fraction of the total weight of jobs arriving. This is in contrast with the speed augmentation model under which no online algorithm for this problem can achieve a competitive ratio independent of p.

Cite as

Anamitra Roy Choudhury, Syamantak Das, and Amit Kumar. Minimizing Weighted lp-Norm of Flow-Time in the Rejection Model. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 25-37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{roychoudhury_et_al:LIPIcs.FSTTCS.2015.25,
  author =	{Roy Choudhury, Anamitra and Das, Syamantak and Kumar, Amit},
  title =	{{Minimizing Weighted lp-Norm of Flow-Time in the Rejection Model}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{25--37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.25},
  URN =		{urn:nbn:de:0030-drops-56341},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.25},
  annote =	{Keywords: online scheduling, flow-time, competitive analysis}
}
Document
Approximating the Regular Graphic TSP in Near Linear Time

Authors: Ashish Chiplunkar and Sundar Vishwanathan

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)


Abstract
We present a randomized approximation algorithm for computing traveling salesperson tours in undirected regular graphs. Given an n-vertex, k-regular graph, the algorithm computes a tour of length at most (1+frac 4+ln 4+varepsilon ln k-O(1)n, with high probability, in O(nk log k) time. This improves upon the result by Vishnoi ([Vishnoi12],FOCS 2012) for the same problem, in terms of both approximation factor, and running time. Furthermore, our result is incomparable with the recent result by Feige, Ravi, and Singh ([FeigeRS14], IPCO 2014), since our algorithm runs in linear time, for any fixed k. The key ingredient of our algorithm is a technique that uses edge-coloring algorithms to sample a cycle cover with O(n/log k) cycles, with high probability, in near linear time. Additionally, we also give a deterministic frac{3}{2}+O(frac{1}sqrt{k}) factor approximation algorithm for the TSP on n-vertex, k-regular graphs running in time O(nk).

Cite as

Ashish Chiplunkar and Sundar Vishwanathan. Approximating the Regular Graphic TSP in Near Linear Time. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 125-135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{chiplunkar_et_al:LIPIcs.FSTTCS.2015.125,
  author =	{Chiplunkar, Ashish and Vishwanathan, Sundar},
  title =	{{Approximating the Regular Graphic TSP in Near Linear Time}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{125--135},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.125},
  URN =		{urn:nbn:de:0030-drops-56264},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.125},
  annote =	{Keywords: traveling salesperson problem, approximation, linear time}
}
Document
Designing Overlapping Networks for Publish-Subscribe Systems

Authors: Jennifer Iglesias, Rajmohan Rajaraman, R. Ravi, and Ravi Sundaram

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
From the publish-subscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and IoT (Internet of Things), new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal is a source or sink of information and builds a physical network in the form of a tree or an overlay network in the form of a star rooted at itself. Every pair of pub-sub terminals that need to be coordinated (e.g. the source and sink of an important piece of control information) define an edge in a bipartite demand graph; the solution must ensure that the corresponding networks rooted at the endpoints of each demand edge overlap at some node. This simple overlap constraint, and the requirement that each network is a tree or a star, leads to a variety of new questions on the design of overlapping networks. In this paper, for the general demand case of the problem, we show that a natural LP formulation has a non-constant integrality gap; on the positive side, we present a logarithmic approximation for the general demand case. When the demand graph is complete, however, we design approximation algorithms with small constant performance ratios, irrespective of whether the pub networks and sub networks are required to be trees or stars.

Cite as

Jennifer Iglesias, Rajmohan Rajaraman, R. Ravi, and Ravi Sundaram. Designing Overlapping Networks for Publish-Subscribe Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 381-395, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{iglesias_et_al:LIPIcs.APPROX-RANDOM.2015.381,
  author =	{Iglesias, Jennifer and Rajaraman, Rajmohan and Ravi, R. and Sundaram, Ravi},
  title =	{{Designing Overlapping Networks for Publish-Subscribe Systems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{381--395},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.381},
  URN =		{urn:nbn:de:0030-drops-53133},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.381},
  annote =	{Keywords: Approximation Algorithms, Steiner Trees, Publish-Subscribe Systems, Integrality Gap, VPN.}
}
Document
A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs

Authors: Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff Stein

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time is a location on the real line. At each integer time, a convex function arrives online. In response, the online algorithm picks a new location. The cost paid by the online algorithm for this response is the distance moved plus the value of the function at the final destination. The objective is then to minimize the aggregate cost over all time. The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem.

Cite as

Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff Stein. A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 96-109, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bansal_et_al:LIPIcs.APPROX-RANDOM.2015.96,
  author =	{Bansal, Nikhil and Gupta, Anupam and Krishnaswamy, Ravishankar and Pruhs, Kirk and Schewior, Kevin and Stein, Cliff},
  title =	{{A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{96--109},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.96},
  URN =		{urn:nbn:de:0030-drops-52970},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.96},
  annote =	{Keywords: Stochastic, Scheduling}
}
Document
Terminal Embeddings

Authors: Michael Elkin, Arnold Filtser, and Ofer Neiman

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
In this paper we study terminal embeddings, in which one is given a finite metric (X,d_X) (or a graph G=(V,E)) and a subset K of X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve approx |K| * |X| pairs, the distortion depends only on |K|, rather than on |X|. We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X \times X and with respect to K * X. Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, Arora et. al. devised an ~O(sqrt(log(r))-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an ~O(sqrt(log |K|)-approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K| <= r, our bound generalizes that of Arora et al.

Cite as

Michael Elkin, Arnold Filtser, and Ofer Neiman. Terminal Embeddings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 242-264, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{elkin_et_al:LIPIcs.APPROX-RANDOM.2015.242,
  author =	{Elkin, Michael and Filtser, Arnold and Neiman, Ofer},
  title =	{{Terminal Embeddings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{242--264},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.242},
  URN =		{urn:nbn:de:0030-drops-53064},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.242},
  annote =	{Keywords: embedding, distortion, terminals}
}
Document
An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem

Authors: Shanfei Li

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


Abstract
In the k-median problem, given a set of locations, the goal is to select a subset of at most k centers so as to minimize the total cost of connecting each location to its nearest center. We study the uniform hard capacitated version of the k-median problem, in which each selected center can only serve a limited number of locations. Inspired by the algorithm of Charikar, Guha, Tardos and Shmoys, we give an improved approximation algorithm for this problem with increasing the capacities by a constant factor, which improves the previous best approximation algorithm proposed by Byrka, Fleszar, Rybicki and Spoerhase.

Cite as

Shanfei Li. An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 325-338, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{li:LIPIcs.APPROX-RANDOM.2014.325,
  author =	{Li, Shanfei},
  title =	{{An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{325--338},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.325},
  URN =		{urn:nbn:de:0030-drops-47062},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.325},
  annote =	{Keywords: Approximation algorithm; k-median problem; LP-rounding; Hard capacities}
}
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