Computable vs Descriptive Combinatorics of Local Problems on Trees

TL;DR

The paper studies locally checkable labeling problems ($LCL$) on $\Delta$-regular trees and situates the computable setting within the broader locality framework by defining descriptive classes $BAIRE$, $CONTINUOUS$, $HCOMP$ and the computable class $COMPUTABLE$. It proves a tight correspondence for highly computable forests: an $LCL$ is in $HCOMP$ iff it is $l$-full for some $l$ (full) using a toast-based construction that yields computable colorings, and otherwise constructs a computable truly $\Delta$-regular forest with no computable $\Pi$-coloring. It then characterizes computable solvability as exactly the greedy $LCL$s, establishing that $COMPUTABLE \subset CONTINUOUS$ with strict inclusion and connecting to continuous coloring results; it also provides results on homomorphism-type $LCL$s and their greedy status. The findings link computable graph coloring to descriptive-set-theoretic and LOCAL-model notions, clarifying when computable algorithms can match Baire-measurable or continuous solutions and illustrating the depth of the divide between highly computable and general computable settings.

Abstract

We study the position of the computable setting in the "common theory of locality" developed in arXiv:2106.02066 and arXiv:2204.09329 for local problems on $Δ$-regular trees, $Δ\in ω$. We show that such a problem admits a computable solution on every highly computable $Δ$-regular forest if and only if it admits a Baire measurable solution on every Borel $Δ$-regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $Δ$ forest then it admits a continuous solution on every maximum degree $Δ$ Borel graph with appropriate topological hypotheses, though the converse does not hold.

Figures (5)

• Figure 1: Stage 0 of the construction for a fixed $t$ for $\Delta = 3$.
• Figure 2: The "uninteresting case" for a step of the construction for a fixed $t$, for $\Delta = 3$.
• Figure 3: The "interesting case" for a step of the construction for a fixed $t$. The top and bottom of the image represent the before and after states respectively. The dashed line encloses one of the new paths, $P_i$.
• Figure 4: An illustration of Definition \ref{['def:greedy']} with $\Delta = 5$ and $k = 3$. True edges are divided with slashes to show the two half edges comprising them. The left hand figure is given, and the right hand figure is the needed extension. The $\alpha$'s are colors from $\Sigma'$, whereas the $\beta$'s can be any elements of $\Sigma$. Definition \ref{['def:greedy']} also technically asks for colors for the other half edges incident to the $y_i$'s, but these do not turn out to be relevant and so are not drawn.
• Figure 5: The "interesting case" for a step of the construction for a fixed $t$ with $\Delta = 3$. The top and bottom image represent the before and after states respectively.

Theorems & Definitions (35)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Theorem 5
• Definition 6
• Theorem 7
• Definition 8
• Theorem 9
• proof