Tree Coloring: Random Order and Predictions

arXiv (Cornell University)

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.

Authors 7

1. Fabian Frei

   Helmholtz Center for Information Security

   Affiliation as printed

   CISPA Helmholtz Center for Information Security , Germany

2. Matthias Gehnen Aachen Department of Computer Science

   RWTH Aachen University

   Affiliation as printed

   Department of Computer Science , RWTH Aachen University , Germany

3. Dennis Komm

   ETH Zurich

   Affiliation as printed

   Department of Computer Science , ETH Zurich , Switzerland

4. Rastislav Kráľovič
5. Richard Královič
6. Peter Rossmanith Aachen Department of Computer Science

   RWTH Aachen University

   Affiliation as printed

   Department of Computer Science , RWTH Aachen University , Germany

7. Moritz Stocker

   ETH Zurich

   Affiliation as printed

   Department of Computer Science , ETH Zurich , Switzerland