[Paper Review] Complexity Thresholds for the Constrained Colored Token Swapping Problem

Davide Bilò, Stefano Leucci+1 more|arXiv (Cornell University)|Jan 14, 2026

Genome Rearrangement Algorithms0 citations

TL;DR

The paper establishes PSPACE-hardness for constrained colored token swapping (CCTS) with four or more colors, even on cubic planar graphs, and shows polynomial-time solvability when the swap graph is a star (hence for k=3). It also presents a general polynomial-time algorithm for CCTS when the swap graph induces a spanning star.

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.

Motivation & Objective

• Determine the computational complexity of the constrained colored token swapping (CCTS) problem across different numbers of colors and swap-graph structures.
• Show PSPACE-hardness for CCTS even on restricted graph classes (planar cubic base graphs) and with limited swap-graph connectivity.
• Provide a polynomial-time algorithm for CCTS when the swap graph is a star (extending to k=3).
• Offer a general framework for deciding CCTS instances when the swap graph forms a spanning star, regardless of k.

Proposed method

• Construct a PSPACE-hardness reduction from the Nondeterministic Constraint Logic (NCL) problem on cubic planar graphs to CCTS, using edge, OR, and AND gadgets that simulate NCL configurations.
• Ensure planarity and cubic base graph by padding gadgets as needed, preserving the reduction’s correctness.
• Prove CCTS is solvable in polynomial time for star swap graphs by adapting the PMG (Pebble Motion on Graphs) framework and defining an equivalence relation on token positions to partition reachable configurations.
• Extend the star-based algorithm to arbitrary k when the admissible swaps induce a spanning star.

Experimental results

Research questions

• RQ1 Is CCTS PSPACE-hard for k ≥ 4 even under restrictive base graph conditions (planar, cubic)?
• RQ2 What is the complexity of CCTS when the number of colors is small (e.g., k=3)?
• RQ3 Can CCTS be solved in polynomial time when the swap graph is a star, irrespective of the base graph structure?
• RQ4 Does a general polynomial-time method exist for instances where the swap graph induces a spanning star for any k?

Key findings

• CCTS is PSPACE-hard even when the base graph is planar and cubic and the swap graph is a path on 4 vertices (k≥4).
• The hardness reduction from NCL uses dedicated edge, OR, and AND gadgets to simulate NCL configurations within CCTS, preserving planarity and degree constraints.
• CCTS is solvable in polynomial time when the swap graph is a star, which extends to k=3 since the star is one of the possible swap graphs for three colors.
• A more general polynomial-time algorithm exists for CCTS when the set of admissible swaps induces a spanning star, yielding polynomial-time solvability for all k in that regime.
• An interactive instance of the constructed reduction is available at the provided URL.
• The results answer open questions about the complexity gap for 3–7 colors left by prior work and unify understanding under star-swap structure.