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.