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DOI: 10.4230/LIPIcs.SoCG.2018.49

Odd Yao-Yao Graphs are Not Spanners

Authors: Yifei Jin, Jian Li, and Wei Zhan

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract

It is a long standing open problem whether Yao-Yao graphs YY_{k} are all spanners [Li et al. 2002]. Bauer and Damian [Bauer and Damian, 2012] showed that all YY_{6k} for k >= 6 are spanners. Li and Zhan [Li and Zhan, 2016] generalized their result and proved that all even Yao-Yao graphs YY_{2k} are spanners (for k >= 42). However, their technique cannot be extended to odd Yao-Yao graphs, and whether they are spanners are still elusive. In this paper, we show that, surprisingly, for any integer k >= 1, there exist odd Yao-Yao graph YY_{2k+1} instances, which are not spanners.

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Yifei Jin, Jian Li, and Wei Zhan. Odd Yao-Yao Graphs are Not Spanners. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jin_et_al:LIPIcs.SoCG.2018.49,
  author =	{Jin, Yifei and Li, Jian and Zhan, Wei},
  title =	{{Odd Yao-Yao Graphs are Not Spanners}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.49},
  URN =		{urn:nbn:de:0030-drops-87621},
  doi =		{10.4230/LIPIcs.SoCG.2018.49},
  annote =	{Keywords: Odd Yao-Yao Graph, Spanner, Counterexample}
}

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DOI: 10.4230/LIPIcs.SWAT.2018.25

SVM via Saddle Point Optimization: New Bounds and Distributed Algorithms

Authors: Lingxiao Huang, Yifei Jin, and Jian Li

Published in: LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Abstract

We study two important SVM variants: hard-margin SVM (for linearly separable cases) and nu-SVM (for linearly non-separable cases). We propose new algorithms from the perspective of saddle point optimization. Our algorithms achieve (1-epsilon)-approximations with running time O~(nd+n sqrt{d / epsilon}) for both variants, where n is the number of points and d is the dimensionality. To the best of our knowledge, the current best algorithm for nu-SVM is based on quadratic programming approach which requires Omega(n^2 d) time in worst case [Joachims, 1998; Platt, 1999]. In the paper, we provide the first nearly linear time algorithm for nu-SVM. The current best algorithm for hard margin SVM achieved by Gilbert algorithm [Gärtner and Jaggi, 2009] requires O(nd / epsilon) time. Our algorithm improves the running time by a factor of sqrt{d}/sqrt{epsilon}. Moreover, our algorithms can be implemented in the distributed settings naturally. We prove that our algorithms require O~(k(d +sqrt{d/epsilon})) communication cost, where k is the number of clients, which almost matches the theoretical lower bound. Numerical experiments support our theory and show that our algorithms converge faster on high dimensional, large and dense data sets, as compared to previous methods.

Cite as

Lingxiao Huang, Yifei Jin, and Jian Li. SVM via Saddle Point Optimization: New Bounds and Distributed Algorithms. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{huang_et_al:LIPIcs.SWAT.2018.25,
  author =	{Huang, Lingxiao and Jin, Yifei and Li, Jian},
  title =	{{SVM via Saddle Point Optimization: New Bounds and Distributed Algorithms}},
  booktitle =	{16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)},
  pages =	{25:1--25:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-068-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{101},
  editor =	{Eppstein, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.25},
  URN =		{urn:nbn:de:0030-drops-88515},
  doi =		{10.4230/LIPIcs.SWAT.2018.25},
  annote =	{Keywords: nu-SVM, hard-margin SVM, saddle point optimization, distributed algorithm}
}

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Documents authored by Jin, Yifei

Odd Yao-Yao Graphs are Not Spanners

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SVM via Saddle Point Optimization: New Bounds and Distributed Algorithms

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