[Paper Review] Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model

Dan Hefetz, Fabian Kühn+2 more|arXiv (Cornell University)|Jul 18, 2016

Complexity and Algorithms in Graphs29 references4 citations

TL;DR

This paper establishes a polynomial lower bound of Ω(Δ^(1/3−η/3)) on the runtime of deterministic distributed graph coloring algorithms in a weak LOCAL model (SET-LOCAL), where nodes receive only sets of messages from neighbors, not individual messages. The result applies to algorithms reducing an O(Δ^{1+η})-coloring to a proper coloring, and the proof leverages recursive neighborhood graph analysis to establish the bound, offering new insight into distributed coloring complexity.

ABSTRACT

We show an $Ω\big(Δ^{\frac{1}{3}-\fracη{3}}\big)$ lower bound on the runtime of any deterministic distributed $\mathcal{O}\big(Δ^{1+η}\big)$-graph coloring algorithm in a weak variant of the \LOCAL\ model. In particular, given a network graph \mbox{$G=(V,E)$}, in the weak \LOCAL\ model nodes communicate in synchronous rounds and they can use unbounded local computation. We assume that the nodes have no identifiers, but that instead, the computation starts with an initial valid vertex coloring. A node can extbf{broadcast} a extbf{single} message of extbf{unbounded} size to its neighbors and receives the extbf{set of messages} sent to it by its neighbors. That is, if two neighbors of a node $v\in V$ send the same message to $v$, $v$ will receive this message only a single time; without any further knowledge, $v$ cannot know whether a received message was sent by only one or more than one neighbor. Neighborhood graphs have been essential in the proof of lower bounds for distributed coloring algorithms, e.g., \cite{linial92,Kuhn2006On}. Our proof analyzes the recursive structure of the neighborhood graph of the respective model to devise an $Ω\big(Δ^{\frac{1}{3}-\fracη{3}}\big)$ lower bound on the runtime for any deterministic distributed $\mathcal{O}\big(Δ^{1+η}\big)$-graph coloring algorithm. Furthermore, we hope that the proof technique improves the understanding of neighborhood graphs in general and that it will help towards finding a lower (runtime) bound for distributed graph coloring in the standard \LOCAL\ model. Our proof technique works for one-round algorithms in the standard \LOCAL\ model and provides a simpler and more intuitive proof for an existing $Ω(Δ^2)$ lower bound.

Motivation & Objective

• To establish a polynomial lower bound on the runtime of deterministic distributed graph coloring algorithms in a weak variant of the LOCAL model.
• To analyze the impact of message aggregation (set reception) on the complexity of graph coloring in distributed systems.
• To provide a novel proof technique based on recursive neighborhood graph structure that may generalize to standard LOCAL model lower bounds.
• To demonstrate that even with unbounded local computation and initial coloring, deterministic coloring remains inherently slow in the SET-LOCAL model.

Proposed method

• Analyzes the recursive structure of neighborhood graphs in the SET-LOCAL model to derive lower bounds on algorithmic runtime.
• Uses a model where nodes receive only sets of messages from neighbors, not individual messages, simulating loss of neighbor identity information.
• Applies combinatorial arguments to show that certain configurations cannot be resolved in fewer than Ω(Δ^(1/3−η/3)) rounds.
• Adapts techniques from prior work on Linial’s algorithm and Kuhn-Wattenhofer color reduction to the SET-LOCAL setting.
• Proves that even one-round algorithms in the standard LOCAL model require Ω(Δ²) colors for d-defective coloring, extending to defective coloring bounds.
• Uses Erdős-type combinatorial results to ensure non-empty sets of non-conflicting colors during color selection.

Experimental results

Research questions

• RQ1 What is the minimum number of rounds required for deterministic distributed graph coloring in the SET-LOCAL model, where nodes only receive message sets from neighbors?
• RQ2 Can the recursive structure of neighborhood graphs be used to derive polynomial lower bounds for coloring algorithms in weak communication models?
• RQ3 How does the absence of neighbor identity information (due to message set aggregation) affect the complexity of distributed coloring?
• RQ4 Does the proof technique extend to provide lower bounds for one-round algorithms in the standard LOCAL model?
• RQ5 What is the minimum number of colors required for a one-round d-defective coloring in the standard LOCAL model?

Key findings

• The paper establishes a lower bound of Ω(Δ^(1/3−η/3)) on the runtime of any deterministic distributed O(Δ^{1+η})-coloring algorithm in the SET-LOCAL model.
• The lower bound holds even when nodes have unbounded local computation and start with an initial valid coloring.
• The proof technique provides a simpler and more intuitive derivation of the known Ω(Δ²) lower bound for one-round coloring in the standard LOCAL model.
• For one-round d-defective coloring in the standard LOCAL model, Ω(Δ²/(d+1)²) colors are required if the input coloring uses sufficiently many colors.
• The Kuhn-Wattenhofer color reduction scheme works without modification in the SET-LOCAL model, as it only requires knowledge of neighbor color sets.
• Linial’s O(log* m)-round O(Δ²)-coloring algorithm is valid in the SET-LOCAL model, as it depends only on message sets, not individual message counts.