Complexity Thresholds for the Constrained Colored Token Swapping Problem

TL;DR

This work studies constrained colored token swapping (CCTS) on graphs, where tokens carry colors and only certain color pairs may swap. It proves a sharp complexity threshold: CCTS is $ \mathsf{PSPACE}$-hard for $k\ge4$, even when the base graph is planar and cubic, via a reduction from Nondeterministic Constraint Logic (NCL) using edge, OR, and AND gadgets; it also shows polynomial-time solvability for $k=3$ and for star swap graphs by a transitive, PMG-based algorithm. The results resolve an open question from Yang and Zhang and delineate a clear boundary in the problem’s complexity, connecting to the Pebble Motion on Graphs and the Friends and Strangers framework. Overall, the paper advances the structural understanding of restricted-token rearrangement problems and identifies tractable and intractable regimes with implications for related puzzles and graph-reconfiguration tasks.

Abstract

Consider the following puzzle: a farmland consists of several fields, each occupied by either a farmer, a fox, a chicken, or a caterpillar. Creatures in neighboring fields can swap positions as long as the fox avoids the farmer, the chicken avoids the fox, and the caterpillar avoids the chicken. The objective is to decide whether there exists a sequence of swaps that rearranges the creatures into a desired final configuration, while avoiding any unwanted encounters. The above puzzle can be cast an instance of the \emph{colored token swapping} problem with $k = 4$ colors (i.e., creature types), in which only certain pairs of colors can be swapped. We prove that such problem is $\mathsf{PSPACE}$-hard even when the graph representing the farmland is planar and cubic. We also show that the problem is polynomial-time solvable when at most three creature types are involved. We do so by providing a more general algorithm deciding instances with arbitrary values of $k$, as long as the set of all admissible swaps between creature types induces a \emph{spanning star}. Our results settle a problem explicitly left open in [Yang and Zhang, IPL 2025], which established $\mathsf{PSPACE}$-completeness for eight creature types and left the complexity status unresolved when the number of creature types is between three and seven.

Figures (10)

• Figure 1: A possible arrangement of fields and creatures of the brain teaser.
• Figure 2: (a), (b), and (c): The initial configuration (a), the final configuration (b) and the swap graph (c) of the CCTS instance corresponding to the brain teaser in Figure \ref{['fig:example']}. (d): The swap graph used in the $\mathsf{PSPACE}$-hardness reduction of CCTS given in YangZ25.
• Figure 3: An edge gadget oriented towards $u$ (a) and towards $v$ (b), enclosed in a gray rectangle. The connection points of the edge gadget are vertices $x_1$ and $x_6$. The connection points $u_i$ and $v_j$ of the gadgets modeling vertices $u$ and $v$ are shown in bold.
• Figure 4: From top left to bottom right: four intermediate configuration encountered when the of color $2$ in Figure \ref{['fig:edge_gadget']} (a) is used to traverse the gadget, causing it to flip its orientation and to end up in the configuration shown in Figure \ref{['fig:edge_gadget']} (b). For the sake of conciseness, not all swaps are shown.
• Figure 5: Three different orientation of the edges incident to an OR node, each with a corresponding configuration of the tokens in the associated OR gadget (enclosed in a gray rectangle). All vertices of the OR gadget are connection points and are shown in bold. The neighboring connection points of the edge gadgets are shown outside the gray rectangles. Only the number of tokens of color $2$ in the gadget is relevant, not the specific vertices on which they are placed.

Theorems & Definitions (8)

• Theorem 1
• Lemma 2
• Lemma 3: KornhauserMS84
• Lemma 4: KornhauserMS84, rephrased
• Lemma 5
• Theorem 6: KornhauserMS84, rephrased
• Lemma 7
• Theorem 8