Paper by Erik D. Demaine
- Zachary Abel, Jeffrey Bosboom, Michael Coulombe, Erik D. Demaine, Linus Hamilton, Adam Hesterberg, Justin Kopinsky, Jayson Lynch, Mikhail Rudoy, and Clemens Thielen, “Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible”, Theoretical Computer Science, volume 839, November 2020, pages 41–102.
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.
- This paper is also available from ScienceDirect and as arXiv:1804.10193.
- The paper is available in PDF (3478k).
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- Related papers:
- Witness_FUN2018 (Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible)
See also other papers by Erik Demaine.
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