3 Search Results for "Lifshitz, Noam"

Document

DOI: 10.4230/LIPIcs.CCC.2024.3

Solving Unique Games over Globally Hypercontractive Graphs

Authors: Mitali Bafna and Dor Minzer

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

We study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders. We show new rounding techniques for higher degree sum-of-squares (SoS) relaxations for worst-case optimization. In particular, our algorithm shows how to round "low-entropy" pseudodistributions, broadly extending the algorithmic framework of [Mitali Bafna et al., 2021]. At a high level, [Mitali Bafna et al., 2021] showed how to round pseudodistributions for problems where there is a "unique" good solution. We extend their framework by exhibiting a rounding for problems where there might be "few good solutions". Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG.

Cite as

Mitali Bafna and Dor Minzer. Solving Unique Games over Globally Hypercontractive Graphs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bafna_et_al:LIPIcs.CCC.2024.3,
  author =	{Bafna, Mitali and Minzer, Dor},
  title =	{{Solving Unique Games over Globally Hypercontractive Graphs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.3},
  URN =		{urn:nbn:de:0030-drops-203996},
  doi =		{10.4230/LIPIcs.CCC.2024.3},
  annote =	{Keywords: unique games, approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2024.9

Quantum Merlin-Arthur and Proofs Without Relative Phase

Authors: Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [Grilo et al., 2014]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu [Jeronimo and Wu, 2023]) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some other constant gap is equal to QMA. One interpretation is that Merlin’s ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) ⊆ NEXP.

Cite as

Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. Quantum Merlin-Arthur and Proofs Without Relative Phase. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bassirian_et_al:LIPIcs.ITCS.2024.9,
  author =	{Bassirian, Roozbeh and Fefferman, Bill and Marwaha, Kunal},
  title =	{{Quantum Merlin-Arthur and Proofs Without Relative Phase}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.9},
  URN =		{urn:nbn:de:0030-drops-195370},
  doi =		{10.4230/LIPIcs.ITCS.2024.9},
  annote =	{Keywords: quantum complexity, QMA(2), PCPs}
}

Document

Extended Abstract

DOI: 10.4230/LIPIcs.ITCS.2021.87

Complexity Measures on the Symmetric Group and Beyond (Extended Abstract)

Authors: Neta Dafni, Yuval Filmus, Noam Lifshitz, Nathan Lindzey, and Marc Vinyals

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huang’s sensitivity theorem using "pseudo-characters", which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size t-intersecting families in the symmetric group and the perfect matching scheme.

Cite as

Neta Dafni, Yuval Filmus, Noam Lifshitz, Nathan Lindzey, and Marc Vinyals. Complexity Measures on the Symmetric Group and Beyond (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 87:1-87:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dafni_et_al:LIPIcs.ITCS.2021.87,
  author =	{Dafni, Neta and Filmus, Yuval and Lifshitz, Noam and Lindzey, Nathan and Vinyals, Marc},
  title =	{{Complexity Measures on the Symmetric Group and Beyond}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{87:1--87:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.87},
  URN =		{urn:nbn:de:0030-drops-136267},
  doi =		{10.4230/LIPIcs.ITCS.2021.87},
  annote =	{Keywords: Computational Complexity Theory, Analysis of Boolean Functions, Complexity Measures, Extremal Combinatorics}
}

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1 Bafna, Mitali
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1 Mathematics of computing → Discrete mathematics
1 Theory of computation → Approximation algorithms analysis
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1 Theory of computation → Quantum complexity theory

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1 Analysis of Boolean Functions
1 Complexity Measures
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