Euclidean distortion and the sparsest cut

Arora, Sanjeev; Lee, James R.; Naor, Assaf

doi:10.1090/s0894-0347-07-00573-5

Cited by 136 publications

(230 citation statements)

References 40 publications

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“…We say that a metric space V = (X, d) is of negative type if there is a map f from X to an Euclidean space such that for all x, y ∈ X we have d(x, y) = f (x) − f (y) 2 2 . (So all triangles spanned by the image of X are acute.)…”

Section: Definitionsmentioning

confidence: 99%

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Hard Metrics from Cayley Graphs of Abelian Groups

Newman

Rabinovich

Stacs 2007

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Hard metrics are the class of extremal metrics with respect to embedding into Euclidean spaces; they incur Ω(log n) multiplicative distortion, which is as large as it can possibly get for any metric space of size n. Besides being very interesting objects akin to expanders and good error-correcting codes, and having a rich structure, such metrics are important for obtaining lower bounds in combinatorial optimization, e. g., on the value of MinCut/MaxFlow ratio for multicommodity flows.For more than a decade, a single family of hard metrics was known (Linial, London, Rabinovich (Combinatorica 1995) and Aumann, Rabani (SICOMP 1998)). Recently, a different family was found by Khot and Naor (FOCS 2005).In this paper we present a general method of constructing hard metrics. Our results extend to embeddings into negative type metric spaces and into 1 .

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Section: Definitionsmentioning

confidence: 99%

“…When studying a special class of metric spaces, perhaps the most natural first question is whether this class contains hard metrics with respect to 2 . Many fundamental results in the modern theory of finite metric spaces may be viewed as a negative answer to this question for some special important class of metrics.…”

Section: Introductionmentioning

confidence: 99%

Hard Metrics from Cayley Graphs of Abelian Groups

Newman

Rabinovich

Stacs 2007

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“…The [26] algorithm is also the only non-trivial combinatorial approximation algorithm for Directed multicut. For more papers on cut problems see [27,6,15,13,35,10]. Some of these papers were an inspiration for this paper.…”

Section: Related Workmentioning

confidence: 99%

On the Advantage of Overlapping Clusters for Minimizing Conductance

2013

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Graph clustering is an important problem with applications to bioinformatics, community discovery in social networks, distributed computing, and more. While most of the research in this area has focused on clustering using disjoint clusters, many real datasets have inherently overlapping clusters. We compare overlapping and non-overlapping clusterings in graphs in the context of minimizing their conductance. It is known that allowing clusters to overlap gives better results in practice. We prove that overlapping clustering may be significantly better than non-overlapping clustering with respect to conductance, even in a theoretical setting.For minimizing the maximum conductance over the clusters, we give examples demonstrating that allowing overlaps can yield significantly better clusterings, namely, one that has much smaller optimum. In addition for the min-max variant, the overlapping version admits a simple approximation algorithm, while our algorithm for the non-overlapping version is complex and yields a worse approximation ratio due to the presence of the additional constraint. Somewhat surprisingly, for the problem of minimizing the sum of conductances, we found out that allowing overlap does not help. We show how to apply a general technique to transform any overlapping clustering into a non-overlapping one with only a modest increase in the sum of conductances. This uncrossing technique is of independent interest and may find further applications in the future.We consider this work as a step toward rigorous comparison of overlapping and nonoverlapping clusterings and hope that it stimulates further research in this area. * IBM T.J.Watson research center.

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“…Clearly, OPT(I) ≤ SDP(I) ≤ α C · OPT(I), and hence the SDP gives an α C approximation to the CSP. 8 The formal statement of Raghavendra's result is:…”

Section: Raghavendra's Resultsmentioning

confidence: 99%

“…In a breakthrough work, Arora, Rao, and Vazirani [10] showed that the SDP algorithm gives O( √ log N ) approximation to the Sparsest Cut problem without demands. This was extended to the demands version of the problem by Arora, Lee, and Naor [8], albeit with a slight loss in the approximation factor. As discussed, the result is equivalent to an upper bound on c 1 (NEG, N ).…”

Section: Theorem 84 ([92 70])mentioning

confidence: 99%

On the Unique Games Conjecture

Khot

46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)

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Euclidean distortion and the sparsest cut

Cited by 136 publications

References 40 publications

Hard Metrics from Cayley Graphs of Abelian Groups

Hard Metrics from Cayley Graphs of Abelian Groups

On the Advantage of Overlapping Clusters for Minimizing Conductance

On the Unique Games Conjecture

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