4 Search Results for "Linial, Nati"

Document
DOI: 10.4230/LIPIcs.CCC.2022.1
The Approximate Degree of Bipartite Perfect Matching

Authors: Gal Beniamini

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the 𝓁_∞-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching of order n, is Θ̃(n^(3/2)). The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small 𝓁₁-norm - only exponential in Θ(n log n). The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube [Aaronson et al., 2021; Huang, 2019]. We show that the spectral sensitivity of perfect matching is exactly Θ(n^(3/2)).
Cite as

Gal Beniamini. The Approximate Degree of Bipartite Perfect Matching. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1:1-1:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{beniamini:LIPIcs.CCC.2022.1,
  author =	{Beniamini, Gal},
  title =	{{The Approximate Degree of Bipartite Perfect Matching}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{1:1--1:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.1},
  URN =		{urn:nbn:de:0030-drops-165634},
  doi =		{10.4230/LIPIcs.CCC.2022.1},
  annote =	{Keywords: Bipartite Perfect Matching, Boolean Functions, Approximate Degree}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.2
An Improved Protocol for the Exactly-N Problem

Authors: Nati Linial and Adi Shraibman

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
In the 3-players exactly-N problem the players need to decide whether x+y+z = N for inputs x,y,z and fixed N. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Even though this is such a basic problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-N problem. This improved protocol has also interesting consequences in additive combinatorics. As we explain below, it yields a higher lower bound on the possible density of corner-free sets in [N]×[N].
Cite as

Nati Linial and Adi Shraibman. An Improved Protocol for the Exactly-N Problem. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 2:1-2:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{linial_et_al:LIPIcs.CCC.2021.2,
  author =	{Linial, Nati and Shraibman, Adi},
  title =	{{An Improved Protocol for the Exactly-N Problem}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{2:1--2:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.2},
  URN =		{urn:nbn:de:0030-drops-142760},
  doi =		{10.4230/LIPIcs.CCC.2021.2},
  annote =	{Keywords: Communication complexity, Number-On-the-Forehead, Corner-free sets}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.54
On the Communication Complexity of High-Dimensional Permutations

Authors: Nati Linial, Toniann Pitassi, and Adi Shraibman

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
We study the multiparty communication complexity of high dimensional permutations in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer n. There is a considerable body of literature dealing with the same problem, where (N,+) is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of research. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that reveal new and unexpected connections between NOF communication complexity of permutations and a variety of well-known problems in combinatorics. We also give a direct algorithmic protocol for Exactly-n. In contrast, all previous constructions relied on large sets of integers without a 3-term arithmetic progression.
Cite as

Nati Linial, Toniann Pitassi, and Adi Shraibman. On the Communication Complexity of High-Dimensional Permutations. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 54:1-54:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{linial_et_al:LIPIcs.ITCS.2019.54,
  author =	{Linial, Nati and Pitassi, Toniann and Shraibman, Adi},
  title =	{{On the Communication Complexity of High-Dimensional Permutations}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{54:1--54:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.54},
  URN =		{urn:nbn:de:0030-drops-101470},
  doi =		{10.4230/LIPIcs.ITCS.2019.54},
  annote =	{Keywords: High dimensional permutations, Number On the Forehead model, Additive combinatorics}
}
Document
DOI: 10.4230/LIPIcs.STACS.2013.526
On the practically interesting instances of MAXCUT

Authors: Yonatan Bilu, Amit Daniely, Nati Linial, and Michael Saks

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract
For many optimization problems, the instances of practical interest often occupy just a tiny part of the algorithm's space of instances. Following (Y. Bilu and N. Linial, 2010), we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, (1 + epsilon)-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time Omega(sqrt(n))-stable instances of MAXCUT, substantially improving the best previously known result.
Cite as

Yonatan Bilu, Amit Daniely, Nati Linial, and Michael Saks. On the practically interesting instances of MAXCUT. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 526-537, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{bilu_et_al:LIPIcs.STACS.2013.526,
  author =	{Bilu, Yonatan and Daniely, Amit and Linial, Nati and Saks, Michael},
  title =	{{On the practically interesting instances of MAXCUT}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{526--537},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.526},
  URN =		{urn:nbn:de:0030-drops-39625},
  doi =		{10.4230/LIPIcs.STACS.2013.526},
  annote =	{Keywords: MAXCUT, Clustering, Hardness in practice, Stability, Non worst-case analysis}
}
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