Who witnesses the witness? Finding witnesses in the witness is hard and sometimes impossible
Author(s)Abel, Zachary R; Bosboom, Jeffrey William; Demaine, Erik D; Hamilton, Linus Ulysses; Hesterberg, Adam Classen; Kopinsky, Justin; Lynch, Jayson R.; Rudoy, Mikhail; ... Show more Show less
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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 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.
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Mathematics
9th International Conference on Fun with Algorithms
Abel, Zachary et al. "Who witnesses the witness? Finding witnesses in the witness is hard and sometimes impossible." 9th International Conference on Fun with Algorithms, June 2018, La Maddalena, Maddalena Islands, Italy, Dagstuhl Research, 2018. © 2018 The Authors
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