Who witnesses the witness? Finding witnesses in the witness is hard and sometimes impossible

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 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.