Tree Coloring: Random Order and Predictions

TL;DR

This paper advances online graph coloring by achieving improved bounds on trees and bipartite graphs in random order and prediction-augmented models, combining theoretical insights with practical algorithms.

Contribution

It introduces new algorithms and bounds for online coloring of trees and bipartite graphs using random order and machine learning predictions.

Findings

Double-logarithmic competitive ratio for trees in random order model

Algorithms utilizing predictions improve coloring performance

Matching lower bounds established for both models

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.