Computational Complexity of Reconfiguring Distance Colorings on Restricted Graph Classes

Core Concepts

Even for planar, bipartite, and 2-degenerate graphs, reconfiguring distance colorings is PSPACE-complete for certain parameter settings.

Abstract

The paper studies the reconfiguration of distance colorings on certain graph classes.
Key highlights:

For d ≥ 2 and k = Ω(d^2), the (d, k)-Coloring Reconfiguration problem is PSPACE-complete even for planar, bipartite, and 2-degenerate graphs. This is shown via a reduction from a variant of the Sliding Tokens problem.
For split graphs, the (2, k)-Coloring Reconfiguration problem is PSPACE-complete for some large k.
The paper provides polynomial-time algorithms to solve the (d, k)-Coloring Reconfiguration problem on graphs of diameter at most d and on paths for any d ≥ 2 and k ≥ d + 1.

The authors first introduce a restricted variant of the Sliding Tokens problem, which they prove to be PSPACE-complete even on planar and 2-degenerate graphs. They then reduce this variant to List (d, k)-Coloring Reconfiguration and further to (d, Ω(d^2))-Coloring Reconfiguration to obtain the main hardness result. For split graphs, they show NP-completeness of the original (2, k)-Coloring problem and extend it to PSPACE-completeness of the reconfiguration variant. Finally, they design polynomial-time algorithms for certain graph classes.

Stats

None.

Quotes

None.

Key Insights Distilled From

Distance Recoloring

by Niranka Bane... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2402.12705.pdf

Deeper Inquiries

What are some practical applications or real-world scenarios where the reconfiguration of distance colorings could be relevant

The reconfiguration of distance colorings can be relevant in various practical applications where constraints on color assignments need to be maintained while allowing for changes. One potential scenario is in wireless sensor networks, where nodes need to be assigned different colors based on their communication range to avoid interference. By reconfiguring the color assignments while ensuring that nodes within a certain distance have different colors, network efficiency and communication reliability can be optimized. Another application could be in scheduling and timetabling problems, where tasks or events need to be assigned colors representing different resources or time slots. Reconfiguring these color assignments while considering distance constraints can help in adapting to changing requirements or constraints in the scheduling process.

Are there any graph classes, beyond the ones considered in this paper, for which the (d, k)-Coloring Reconfiguration problem could be solvable in polynomial time

While the paper discusses the complexity of (d, k)-Coloring Reconfiguration on planar, bipartite, and 2-degenerate graphs, there may be other graph classes where the problem could be solvable in polynomial time. For example, on certain tree structures or sparse graph classes with specific properties, the reconfiguration of distance colorings might have efficient solutions. Additionally, for graphs with special structures like interval graphs, chordal graphs, or perfect graphs, the problem could potentially be tractable due to the inherent properties of these graph classes that simplify the reconfiguration process. Further research could explore these graph classes to identify conditions under which the (d, k)-Coloring Reconfiguration problem becomes polynomial-time solvable.

How could the techniques developed in this paper be extended to study the reconfiguration of other generalized coloring problems, such as list coloring or list distance coloring

The techniques developed in this paper for studying the reconfiguration of distance colorings could be extended to investigate the reconfiguration of other generalized coloring problems, such as list coloring or list distance coloring. By adapting the reduction methods and complexity analysis frameworks used in this study, researchers could explore the reconfiguration complexities of list coloring problems on different graph classes. Additionally, the concept of forbidding paths and admissible pairs introduced in the context of distance colorings could be applied to study the reconfiguration of list colorings with additional constraints. This extension could provide insights into the computational complexity of reconfiguring list colorings on various graph structures and help in understanding the inherent challenges of dynamic color assignment problems.