Diameter-Free Distributed Frequency Control for Graph Coloring in the CONGEST Model

2026-08-03

AI Summary

This work addresses the problem of achieving proper graph coloring with balanced color frequencies in the synchronous CONGEST model, without relying on the network diameter. The authors propose two randomized distributed algorithms: the first achieves an approximately uniform color distribution within $O((\lambda+1)\log n)$ rounds with high probability; the second, using $\Delta + \lceil(\Delta+1)/\lceil\ln n\rceil\rceil$ colors, guarantees that each color class has size at most $O((\lambda+1)(\sigma \log^2 n + \log n))$. This is the first scheme to provide one-sided or two-sided guarantees on color frequency balance without any dependence on the graph diameter, introducing a novel mechanism based on palette slackness to bound color usage frequencies.

Abstract

This paper presents two randomized proper-coloring algorithms that control color frequencies in the synchronous CONGEST model without paying a diameter-dependent coordination cost. Let $λ\geq 1$ denote the desired failure exponent. For every fixed $δ> 0$, the first algorithm uses $χ= \lceil (2+δ)Δ\rceil$ colors and, with probability at least $1 - n^{-λ}$, outputs a proper coloring that bounds the deviation of every color frequency from $n/χ$ by $O_δ(\sqrt{(λ+1)(n/χ)\lg n} + (λ+1)\lg n)$. Under an explicit load condition, this additive guarantee yields two-sided relative balance. The second algorithm works with every $χ> Δ$ and gives a one-sided frequency cap controlled by the palette slack $χ- Δ$. In particular, it uses $Δ+ \lceil (Δ+1)/\lceil \ln n \rceil \rceil$ colors and caps every used color class by $O((λ+1)(σ\lg^2 n + \lg n))$, where $σ= n/(Δ+1)$. Both algorithms run in $O((λ+1)\lg n)$ rounds, with no dependence on the network diameter; for the first algorithm, the multiplicative constant in the time bound depends on $δ$.

Problem

• distributed graph coloring
• frequency control
• CONGEST model
• diameter-free
• color balance

Innovation

• diameter-free
• distributed graph coloring
• frequency control
• CONGEST model
• randomized algorithm