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Tree Coloring: Random Order and Predictions

Fabian Frei, Matthias Gehnen, Dennis Komm, Rastislav Královič, Richard Královič, Peter Rossmanith, Moritz Stocker

TL;DR

This work studies online graph coloring under two relaxed models: random-order vertex arrivals and online coloring with predictions. It shows that FirstFit achieves $O\left(\dfrac{\log n}{\log\log n}\right)$ colors in expectation on trees in the random-order model, using probabilistic tail bounds and Stirling approximations to bound color usage. In the predictions model, algorithms with advice achieve strong, robust guarantees (consistency with perfect advice and bounded degradation with errors) and extend to all bipartite graphs, with matching lower bounds. Finally, the paper demonstrates a combined approach where adversarial instances are presented in random order with predictions, yielding further improvements and reducing the pessimism of classical online coloring. The results provide a cohesive framework for leveraging randomness and predictions to dramatically improve online coloring of bipartite graphs and trees, with implications for practical coloring under limited adversarial power.

Abstract

Coloring is a notoriously hard problem, and even more so in the online setting, where each arriving vertex has to be colored immediately and irrevocably. Already on trees, which are trivially two-colorable, it is impossible to achieve anything better than a logarithmic competitive ratio. We show how to undercut this bound by a double-logarithmic factor in the slightly relaxed online model where the vertices arrive in random order. We then also analyze algorithms with predictions, showing how well we can color trees with machine-learned advice of varying reliability. We further extend our analysis to all two-colorable graphs and provide matching lower bounds in both cases. Finally, we demonstrate how the two mentioned approaches, both of which diminish the often unjustified pessimism of the classical online model, can be combined to yield even better results.

Tree Coloring: Random Order and Predictions

TL;DR

This work studies online graph coloring under two relaxed models: random-order vertex arrivals and online coloring with predictions. It shows that FirstFit achieves colors in expectation on trees in the random-order model, using probabilistic tail bounds and Stirling approximations to bound color usage. In the predictions model, algorithms with advice achieve strong, robust guarantees (consistency with perfect advice and bounded degradation with errors) and extend to all bipartite graphs, with matching lower bounds. Finally, the paper demonstrates a combined approach where adversarial instances are presented in random order with predictions, yielding further improvements and reducing the pessimism of classical online coloring. The results provide a cohesive framework for leveraging randomness and predictions to dramatically improve online coloring of bipartite graphs and trees, with implications for practical coloring under limited adversarial power.

Abstract

Coloring is a notoriously hard problem, and even more so in the online setting, where each arriving vertex has to be colored immediately and irrevocably. Already on trees, which are trivially two-colorable, it is impossible to achieve anything better than a logarithmic competitive ratio. We show how to undercut this bound by a double-logarithmic factor in the slightly relaxed online model where the vertices arrive in random order. We then also analyze algorithms with predictions, showing how well we can color trees with machine-learned advice of varying reliability. We further extend our analysis to all two-colorable graphs and provide matching lower bounds in both cases. Finally, we demonstrate how the two mentioned approaches, both of which diminish the often unjustified pessimism of the classical online model, can be combined to yield even better results.
Paper Structure (2 sections, 1 theorem, 1 equation)

This paper contains 2 sections, 1 theorem, 1 equation.

Table of Contents

  1. Introduction
  2. The Random-Order Model

Key Result

Lemma 1

Let $T=(V,E)$ be any tree. Let $v\in V$ be a vertex of $T$ and let $c(v)$ be the color assigned to $v$ by $\textup{FirstFit}$. Then

Theorems & Definitions (1)

  • Lemma 1