A Tight Lower Bound for 3-Coloring Grids in the Online-LOCAL Model

TL;DR

The paper analyzes the locality of graph coloring problems in the deterministic Online-LOCAL model, revealing tight bounds across several graph classes. It introduces a novel b-value framework to prove an $Ω(log n)$ lower bound for 3-coloring in grids, and extends to a stronger $Ω(√n)$ bound on toroidal/cylindrical grids. It then constructs gadget-based hard instances to show an $Ω(n)$ locality lower bound for $(2k-2)$-coloring in $k$-partite graphs (for $k≥3$), while identifying a broad class of graphs $L_{k,ℓ}$ with locally inferable unique colorings that admit an $O(log n)$-local Online-LOCAL algorithm for $(k+1)$-coloring, with matching lower bounds. The results demonstrate a nuanced landscape where model differences (especially global memory in Online-LOCAL) lead to substantial locality gaps, and they connect to recent LOCAL-model lower bounds. Collectively, the work sharpens our understanding of locality for fundamental coloring problems and highlights algorithmic techniques that achieve near-optimal performance on structured graph families.

Abstract

Recently, \citeauthor*{akbari2021locality}~(ICALP 2023) studied the locality of graph problems in distributed, sequential, dynamic, and online settings from a {unified} point of view. They designed a novel $O(\log n)$-locality deterministic algorithm for proper 3-coloring bipartite graphs in the $\mathsf{Online}$-$\mathsf{LOCAL}$ model. In this work, we establish the optimality of the algorithm by showing a \textit{tight} deterministic $Ω(\log n)$ locality lower bound, which holds even on grids. To complement this result, we have the following additional results: \begin{enumerate} \item We show a higher and {tight} $Ω(\sqrt{n})$ lower bound for 3-coloring toroidal and cylindrical grids. \item Considering the generalization of $3$-coloring bipartite graphs to $(k+1)$-coloring $k$-partite graphs, %where $k \geq 2$ is a constant, we show that the problem also has $O(\log n)$ locality when the input is a $k$-partite graph that admits a \emph{locally inferable unique coloring}. This special class of $k$-partite graphs covers several fundamental graph classes such as $k$-trees and triangular grids. Moreover, for this special class of graphs, we show a {tight} $Ω(\log n)$ locality lower bound. \item For general $k$-partite graphs with $k \geq 3$, we prove that the problem of $(2k-2)$-coloring $k$-partite graphs exhibits a locality of $Ω(n)$ in the $\onlineLOCAL$ model, matching the round complexity of the same problem in the $\LOCAL$ model recently shown by \citeauthor*{coiteux2023no}~(STOC 2024). Consequently, the problem of $(k+1)$-coloring $k$-partite graphs admits a locality lower bound of $Ω(n)$ when $k\geq 3$, contrasting sharply with the $Θ(\log n)$ locality for the case of $k=2$. \end{enumerate}

Figures (8)

• Figure 1: The unique tripartition of a triangular grid restricted to a connected subgraph.
• Figure 2: Cancellation of $a$-values.
• Figure 3: A directed path whose $b$-value is zero.
• Figure 4: Intersection of a directed cycle and a region of nodes with colors $1$ and $2$.
• Figure 5: Concatenating $P_{u,v}$ and $P_{s,t}$.

Theorems & Definitions (53)

• Theorem 1
• Corollary 1.1
• Theorem 2
• Corollary 1.2
• Theorem 3
• Corollary 1.3
• Definition 1.4: Locally inferable unique colorings
• Theorem 4
• Corollary 1.5
• Theorem 5