Optimal Online Edge Coloring of Planar Graphs with Advice

Abstract

AbstractUsing 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.KeywordsBipartite GraphPlanar GraphMaximum DegreeCompetitive RatioOnline AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.