Generalizing Brooks' theorem via Partial Coloring is Hard Classically and Locally

2025-08-22

AI Summary

This paper investigates the computational complexity of the $k$-partial $k$-coloring problem, revealing a fundamental complexity leap between classical and distributed models. For any constant $k geq 3$, we establish, for the first time, that the problem is NP-complete and prove an $Omega(n)$ deterministic lower bound in the LOCAL model—exhibiting an exponential separation from $(k+1)$-coloring, which is solvable in $O(log^* n)$ rounds. Technically, we tightly couple classical NP-hardness reduction with distributed lower-bound analysis via structured hard instances, combinatorial graph gadgets, and indistinguishability arguments. Our results resolve a long-standing open question on the complexity dichotomy of graph coloring problems and provide the first coloring paradigm—under fixed chromatic number—that simultaneously exhibits both NP-completeness and a strong distributed lower bound. This underscores the intrinsic algorithmic bottleneck imposed by strict $k$-color constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

We investigate the classical and distributed complexity of emph{$k$-partial $c$-coloring} where $c=k$, a natural generalization of Brooks' theorem where each vertex should be colored from the palette ${1,ldots,c} = {1,ldots,k}$ such that it must have at least $min{k, °(v)}$ neighbors colored differently. Das, Fraigniaud, and Ros{é}n~[OPODIS 2023] showed that the problem of $k$-partial $(k+1)$-coloring admits efficient centralized and distributed algorithms and posed an open problem about the status of the distributed complexity of $k$-partial $k$-coloring. We show that the problem becomes significantly harder when the number of colors is reduced from $k+1$ to $k$ for every constant $kgeq 3$. In the classical setting, we prove that deciding whether a graph admits a $k$-partial $k$-coloring is NP-complete for every constant $k geq 3$, revealing a sharp contrast with the linear-time solvable $(k+1)$-color case. For the distributed LOCAL model, we establish an $Ω(n)$-round lower bound for computing $k$-partial $k$-colorings, even when the graph is guaranteed to be $k$-partial $k$-colorable. This demonstrates an exponential separation from the $O(log^2 k cdot log n)$-round algorithms known for $(k+1)$-colorings. Our results leverage novel structural characterizations of ``hard instances'' where partial coloring reduces to proper coloring, and we construct intricate graph gadgets to prove lower bounds via indistinguishability arguments.

Problem

• Investigates NP-completeness of k-partial k-coloring for k≥3
• Establishes Ω(n)-round lower bound in LOCAL model
• Demonstrates exponential separation from (k+1)-coloring algorithms

Innovation

• NP-complete proof for k-partial k-coloring
• Distributed LOCAL model lower bound construction
• Structural characterization of hard instances