Complexity Thresholds for the Constrained Colored Token Swapping Problem

2026-01-14

AI Summary

This study investigates the computational complexity of the constrained colored token swapping problem: given initial and target configurations, determine whether a valid sequence of adjacent swaps exists, where only tokens of specific types are permitted to swap. By employing graph-theoretic modeling, complexity reductions, and combinatorial algorithm design, the work establishes that the problem is PSPACE-hard for four token types—even when restricted to planar cubic graphs—thereby closing the complexity gap between three and seven types. Additionally, the paper presents a polynomial-time algorithm for the case of three token types when the underlying swap graph forms a generating star, enabling efficient solvability under this structural constraint.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

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.

Problem

• colored token swapping
• computational complexity
• PSPACE-hardness
• constrained swaps
• graph reconfiguration

Innovation

• constrained colored token swapping
• PSPACE-hardness
• spanning star
• planar cubic graphs
• polynomial-time algorithm