2014/10/29 by Jesper W. Mikkelsen, Mikkelsen, Jesper W.
Computer Science · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Optimization and Search Problems #cs.DS
paper · pdf · doi:10.48550/arxiv.1410.7923
CIAC 2015
openalex publication_date 2014/10/29 · arxiv created 2015/02/12 · arxiv updated 2015/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Δ, it follows from Vizing's Theorem that O(mlog Δ) 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 Δ is. On the other hand, we show that Ω(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 Ω(mlog Δ) bits of advice to achieve optimality.