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Documents authored by Hamilton, Linus

Document
DOI: 10.4230/LIPIcs.ITCS.2019.41
The Paulsen Problem Made Simple

Authors: Linus Hamilton and Ankur Moitra

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

Abstract
The Paulsen problem is a basic problem in operator theory that was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every epsilon-nearly equal norm Parseval frame in d dimensions is within squared distance O(epsilon d^{13/2}) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O(epsilon d^2).
Cite as

Linus Hamilton and Ankur Moitra. The Paulsen Problem Made Simple. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 41:1-41:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hamilton_et_al:LIPIcs.ITCS.2019.41,
  author =	{Hamilton, Linus and Moitra, Ankur},
  title =	{{The Paulsen Problem Made Simple}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{41:1--41:6},
  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.2019.41},
  URN =		{urn:nbn:de:0030-drops-101347},
  doi =		{10.4230/LIPIcs.ITCS.2019.41},
  annote =	{Keywords: radial isotropic position, operator scaling, Paulsen problem}
}
Document
DOI: 10.4230/LIPIcs.FUN.2018.3
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

Authors: Zachary Abel, Jeffrey Bosboom, Erik D. Demaine, Linus Hamilton, Adam Hesterberg, Justin Kopinsky, Jayson Lynch, and Mikhail Rudoy

Published in: LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Abstract
We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies.
Cite as

Zachary Abel, Jeffrey Bosboom, Erik D. Demaine, Linus Hamilton, Adam Hesterberg, Justin Kopinsky, Jayson Lynch, and Mikhail Rudoy. Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 3:1-3:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abel_et_al:LIPIcs.FUN.2018.3,
  author =	{Abel, Zachary and Bosboom, Jeffrey and Demaine, Erik D. and Hamilton, Linus and Hesterberg, Adam and Kopinsky, Justin and Lynch, Jayson and Rudoy, Mikhail},
  title =	{{Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible}},
  booktitle =	{9th International Conference on Fun with Algorithms (FUN 2018)},
  pages =	{3:1--3:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-067-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{100},
  editor =	{Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.3},
  URN =		{urn:nbn:de:0030-drops-87944},
  doi =		{10.4230/LIPIcs.FUN.2018.3},
  annote =	{Keywords: video games, puzzles, hardness}
}
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