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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, Mikhail Rudoy



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Author Details

Zachary Abel
  • MIT EECS Department, 50 Vassar St., Cambridge, MA 02139, USA
Jeffrey Bosboom
  • MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
Erik D. Demaine
  • MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
Linus Hamilton
  • MIT Mathematics Department, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Adam Hesterberg
  • MIT Mathematics Department, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Justin Kopinsky
  • MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
Jayson Lynch
  • MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
Mikhail Rudoy
  • MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA

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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)
https://doi.org/10.4230/LIPIcs.FUN.2018.3

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • video games
  • puzzles
  • hardness

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References

  1. 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. arXiv:1804.10193, 2018. Google Scholar
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