Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
Name
1804.10193.pdf
Description
Submitted version
Size
3.63 MB
Format
Adobe PDF
Checksum (MD5)
1a90368ef15f6d72c9ab1b70e9e21600
Author(s)
Abel, Zachary R
Bosboom, Jeffrey William
Coulombe, Michael Joseph
Demaine, Erik D.
Hamilton, Linus
Hesterberg, Adam
Kopinsky, Justin
Lynch, Jayson R.
Rudoy, Mikhail
Thielen, Clemens
Date Issued
June 2020
Journal
Theoretical Computer Science
Publisher
Elsevier BV
Citation
Abel, Zachary et al. "Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible." Theoretical Computer Science 839 (November 2020): 41-102
Version
Original manuscript
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 simple 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 Σ2-complete (“witnesses do not exist”), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving “subset Hamiltonian path” for terminals on the boundary of an embedded planar graph in polynomial time.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.tcs.2020.05.031
