Complexity Thresholds for the Constrained Colored Token Swapping Problem

TL;DR

This paper investigates the complexity of a constrained colored token swapping problem, proving it is PSPACE-hard for four or more creature types on planar cubic graphs, but polynomial-time solvable with up to three types.

Contribution

It establishes the complexity thresholds for the problem, resolving an open question about the complexity for three to seven creature types and providing a general algorithm for certain swap constraints.

Findings

PSPACE-hardness for four or more creature types on planar cubic graphs

Polynomial-time solvability for up to three creature types

A general algorithm for instances with spanning star swap sets

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.