Branching Width when Solving Random Jigsaw Puzzles with a Planted Solution

John Tsotsos asked his AI students to work on having an AI solve a jigsaw puzzle, though it is NP-complete. Instead of looking at a real puzzle, this paper considers a randomly generated puzzle with a planted solution whose probability of piece connection is set so that each piece connects to Θ( √ ǫn) pieces but with 1 − Θ( ǫ √ n ) probability the planted solution is unique. As expected, the recursive backtracking algorithm, which attempts to put together just the edge pieces into a frame, branches exponentially as does building a triangle from a corner of the puzzle. Building a block in the middle does better, but still branches super (quasi) polynomially. In contrast, expanding a rectangle from a corner takes only linear expected time (or quadratic depending on the model). In fact, the expected number of backtracking branches alive at any given time is 1+Θ(ǫ). It is also interesting that for the different orders of completing the puzzle, there are very different “reasons” for the exponential blowup. We coded the algorithm and it ran just as described.