Optimally Solving Colored Generalized Sliding-Tile Puzzles: Complexity and Bounds

TL;DR

This study establishes the computational complexity of CGSP and its key sub-problems under a broad spectrum of possible conditions and characterizes solution makespan lower and upper bounds that differ by at most a logarithmic factor.

Abstract

The Generalized Sliding-Tile Puzzle (GSTP), allowing many square tiles on a board to move in parallel while enforcing natural geometric collision constraints on the movement of neighboring tiles, provide a high-fidelity mathematical model for many high-utility existing and future multi-robot applications, e.g., at mobile robot-based warehouses or autonomous garages. Motivated by practical relevance, this work examines a further generalization of GSTP called the Colored Generalized Sliding-Tile Puzzle (CGSP), where tiles can now assume varying degrees of distinguishability, a common occurrence in the aforementioned applications. Our study establishes the computational complexity of CGSP and its key sub-problems under a broad spectrum of possible conditions and characterizes solution makespan lower and upper bounds that differ by at most a logarithmic factor. These results are further extended to higher-dimensional versions of the puzzle game.

Figures (5)

• Figure 1: [left] Start and goal configurations of a $15$-puzzle instance. In the generalized sliding-tile puzzle, modeled after the $15$-puzzle, there can be $1+$ escorts and multiple tiles may move synchronously, e.g., tile $3$ and $9$ may move to the right in a single step in the left configuration. [right] In a Klotski puzzle (bearing many other names such as Huarong Road) WikiKlotski, tiles are rearranged via sliding to allow the large square red title to "escape" from the green opening. Whereas the goal configuration in the $15$-puzzle is a single fixed one, the goal configuration in the Klotski only specifies the location of the largest tile and leaves the other tiles' final configuration unspecified.
• Figure 2: Illustration of escort teleportation in a BGSP instance. The arrows show the teleportation intents, which are executed in a single step by moving tiles synchronously in opposite directions.
• Figure 3: Illustration of the MOBGSP instance constructed from a 3SAT instance, showing the initial configuration. $G$ corresponds to the smallest rectangular region enclosing all the black tiles aside from the top row. $H$, the SCG board, is highlighted in orange. Not drawn to scale.
• Figure 4: Illustration of handling multi-escort MOBGSP instances by adding additional independent escort jobs. Only the lower part of the overall grid is shown, with the upper part truncated. Not drawn to scale.
• Figure 5: Illustration of the algorithm on an $8 \times 8$ instance, starting with $4 \times 4$ squares. From a starting configuration (1), the tiles are moved into the inner squares to reach (2) as in the 2nd stage. Then the squares are merged in matching pairs in the horizontal direction from (2)--(5) as in the 3rd stage, where blue tiles in the even column group are brought down to get (3), followed by highway rectangular shifts to move them underneath the matching square as in (4) to be subsequently absorbed as in (5). The same procedure is applied in the vertical direction from (5)--(8) to get a nearly sorted configuration, which is then brought down to the bottom row as in the 4th stage.

Theorems & Definitions (27)

• remark thmcounterremark
• remark thmcounterremark
• remark thmcounterremark
• lemma thmcounterlemma
• lemma thmcounterlemma
• lemma thmcounterlemma
• theorem thmcountertheorem
• corollary thmcountercorollary
• corollary thmcountercorollary
• proof