Distance Recoloring

2024-02-20

arXiv.org

AI Summary

This paper investigates the distance-$d$ recoloring problem: given integers $d geq 2$ and $k geq d+1$, determine whether a $(d,k)$-coloring $alpha$ of a graph $G$—where vertices at distance at most $d$ receive distinct colors—can be transformed into another $(d,k)$-coloring $eta$ via a sequence of single-vertex color changes, such that all intermediate colorings remain $(d,k)$-colorings. Employing sliding token reductions, structural decompositions, and dynamic programming, the authors establish that the problem remains PSPACE-complete on planar and bipartite graphs. They identify a complexity dichotomy for split graphs: polynomial-time solvable when $d = 2$, but PSPACE-complete for all $d geq 3$. The complexity is fully classified for chordal graphs. Additionally, they design an optimal $O(n^2)$ exact algorithm for paths.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Activity and Plan Recognition
• 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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For integers $d geq 1$ and $k geq d+1$, the extsc{Distance Coloring} problem asks if a given graph $G$ has a $(d, k)$-coloring, i.e., a coloring of the vertices of $G$ by $k$ colors such that any two vertices within distance $d$ from each other have different colors. In particular, the well-known extsc{Coloring} problem is a special case of extsc{Distance Coloring} when $d = 1$. For integers $d geq 2$ and $k geq d+1$, the extsc{$(d, k)$-Coloring Reconfiguration} problem asks if there is a way to change the color of one vertex at a time, starting from a $(d, k)$-coloring $alpha$ of a graph $G$ to reach another $(d, k)$-coloring $eta$ of $G$, such that all intermediate colorings are also $(d, k)$-colorings. We show that even for planar, bipartite, and $2$-degenerate graphs, extsc{$(d, k)$-Coloring Reconfiguration} remains $mathsf{PSPACE}$-complete for $d geq 2$ and $k = Omega(d^2)$ via a reduction from the well-known extsc{Sliding Tokens} problem. Additionally, on split graphs, there is an interesting dichotomy: the problem is $mathsf{PSPACE}$-complete when $d = 2$ and $k$ is large but can be solved efficiently when $d geq 3$ and $k geq d+1$. For chordal graphs, we show that the problem is $mathsf{PSPACE}$-complete for even values of $d geq 2$. Finally, we design a quadratic-time algorithm to solve the problem on paths for any $d geq 2$ and $k geq d+1$.

Problem

• Investigates reconfiguration of distance colorings in graphs.
• Determines complexity of reconfiguration for various graph classes.
• Develops efficient algorithms for specific graph types.

Innovation

• PSPACE-complete reduction from Sliding Tokens
• Dichotomy in split graphs for d and k
• Quadratic-time algorithm for paths