Online Coloring and a New Type of Adversary for Online Graph Problems

Abstract

We introduce a new type of adversary for online graph problems thus allowing parametrized analysis of online algorithms with respect to a natural parameter. The new adversary is parameterized by a single integer \(\kappa \), which upper bounds the number of connected components that the adversary can use at any time during the presentation of the online graph G. We call this adversary “\(\kappa \)-components-bounded”, or \(\kappa \)-CB for short. On one hand, this adversary is restricted compared to the classical adversary because of the \(\kappa \)-CB constraint. On the other hand, we seek competitive ratios parameterized only by \(\kappa \) with no dependence on the input length n, thereby giving the new adversary power to use arbitrarily large inputs. We study online coloring under the \(\kappa \)-CB adversary. We obtain a finer analysis of the existing algorithms FirstFit and CBIP by computing their competitive ratios on trees and bipartite graphs under the new adversary: (1) Perhaps surprisingly, FirstFit outperforms CBIP on trees; (2) The competitive ratio of CBIP on bipartite graphs is simply \(\kappa \). We also study several well known classes of graphs, such as 3-colorable, \(C_k\)-free, d-inductive, planar, and bounded treewidth, with respect to online coloring under the \(\kappa \)-CB adversary. We demonstrate that the extra adversarial power of unbounded input length outweighs the restriction on the number of connected components leading to non-existence of competitive algorithms for these classes.