Optimal Online Edge Coloring of Planar Graphs with Advice

TL;DR

This paper investigates the amount of future knowledge needed for online edge coloring of graphs, showing that bounded degeneracy graphs require linear advice, while bipartite graphs need advice proportional to their maximum degree for optimal coloring.

Contribution

It establishes bounds on advice complexity for optimal online edge coloring in various graph classes, including bounded degeneracy and bipartite graphs.

Findings

Bounded degeneracy graphs need only O(m) advice bits for optimal coloring.

Achieving better than no-advice competitive ratio requires Ω(m) advice bits.

Bipartite graphs require Ω(m log Δ) advice bits for optimal coloring with fixed advice per edge.

Abstract

Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree $\Delta$, it follows from Vizing's Theorem that $O(m\log \Delta)$ bits of advice suffice to achieve optimality, where $m$ is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $\Delta$ is. On the other hand, we show that $\Omega (m)$ bits of advice are necessary just to achieve a competitive ratio better than that of the best deterministic online algorithm without advice. Furthermore, we consider algorithms which use a fixed number of advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to this class of algorithms). We show that for bipartite graphs, any such algorithm must use at least $\Omega(m\log \Delta)$ bits of advice to achieve optimality.