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DOI: 10.4230/LIPIcs.ITCS.2022.60

Multiscale Entropic Regularization for MTS on General Metric Spaces

Authors: Farzam Ebrahimnejad and James R. Lee

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

We present an O((log n)²)-competitive algorithm for metrical task systems (MTS) on any n-point metric space that is also 1-competitive for service costs. This matches the competitive ratio achieved by Bubeck, Cohen, Lee, and Lee (2019) and the refined competitive ratios obtained by Coester and Lee (2019). Those algorithms work by first randomly embedding the metric space into an ultrametric and then solving MTS there. In contrast, our algorithm is cast as regularized gradient descent where the regularizer is a multiscale metric entropy defined directly on the metric space. This answers an open question of Bubeck (Highlights of Algorithms, 2019).

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Farzam Ebrahimnejad and James R. Lee. Multiscale Entropic Regularization for MTS on General Metric Spaces. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 60:1-60:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ebrahimnejad_et_al:LIPIcs.ITCS.2022.60,
  author =	{Ebrahimnejad, Farzam and Lee, James R.},
  title =	{{Multiscale Entropic Regularization for MTS on General Metric Spaces}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{60:1--60:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.60},
  URN =		{urn:nbn:de:0030-drops-156568},
  doi =		{10.4230/LIPIcs.ITCS.2022.60},
  annote =	{Keywords: Metrical task systems, online algorithms, metric embeddings, convex optimization}
}

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DOI: 10.4230/LIPIcs.ITCS.2022.61

Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs

Authors: Farzam Ebrahimnejad, Ansh Nagda, and Shayan Oveis Gharan

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

We show that the ratio of the number of near perfect matchings to the number of perfect matchings in d-regular strong expander (non-bipartite) graphs, with 2n vertices, is a polynomial in n, thus the Jerrum and Sinclair Markov chain [Jerrum and Sinclair, 1989] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least Ω(d)ⁿ many perfect matchings, thus proving the Lovasz-Plummer conjecture [L. Lovász and M.D. Plummer, 1986] for this family of graphs.

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Farzam Ebrahimnejad, Ansh Nagda, and Shayan Oveis Gharan. Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ebrahimnejad_et_al:LIPIcs.ITCS.2022.61,
  author =	{Ebrahimnejad, Farzam and Nagda, Ansh and Gharan, Shayan Oveis},
  title =	{{Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{61:1--61:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.61},
  URN =		{urn:nbn:de:0030-drops-156579},
  doi =		{10.4230/LIPIcs.ITCS.2022.61},
  annote =	{Keywords: perfect matchings, approximate sampling, approximate counting, expanders}
}

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Documents authored by Ebrahimnejad, Farzam

Multiscale Entropic Regularization for MTS on General Metric Spaces

Abstract

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Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs

Abstract

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