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 ${\mathrm{\Sigma }}_2$-complete (“witnesses do not exist”), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. 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.

我们分析了2016年益智视频游戏《见证》中精选的多种铅笔和纸质拼图的计算复杂性。在所有难题中，目标都是在矩形网格图中绘制从起始顶点到目标顶点的简单路径。不同的拼图类型在路径上设置了不同的约束：防止访问某些边缘（折边）；强迫一些边缘或顶点被访问（六边形）；迫使某些单元具有一定数量的入射路径边缘（三角形）；或强制由路径形成的区域是部分单色的（正方形），正好具有两个特殊的单元格（星形），或者被给定形状（多米诺骨牌）和/或负计数形状（反多米诺骨牌）单独覆盖。我们证明任何一个这些线索类型（第一个除外）中的任何一个都足以使路径查找NP完整（“见证人存在，但很难找到”），即使对于矩形板也是如此。此外，我们显示出最终线索类型（抗体）可以“寻找”同一区域中另一个线索的作用，因此可以找到路径${\mathrm{\Sigma }}_2$-完整（“见证人不存在”），即使仅使用一种抗体（与许多抗/多氨基酸）结合使用，使用多种抗体也不会使问题变得更加棘手。从积极的一面，我们给出了一种单项线索的多项式时间算法，方法是将难题边界上的线索减少为六边形线索，即使存在断边的情况下，也可以求解边界上的“子集哈密顿路径”在多项式时间内嵌入平面图。