Lower bounds for on-line interval coloring with vector and cardinality constraints

Abstract

We propose two strategies for Presenter in the on-line interval graph coloring games. Specifically, we consider a setting in which each interval is associated with a d-dimensional vector of weights and the coloring needs to satisfy the d-dimensional bandwidth constraint, and the k-cardinality constraint. Such a variant was first introduced by Epstein and Levy and it is a natural model for resource-aware task scheduling with d different shared resources where at most k tasks can be scheduled simultaneously on a single machine. The first strategy forces any on-line interval coloring algorithm to use at least (Formula presented) different colors on an (Formula presented)- colorable set of intervals. The second strategy forces any on-line interval coloring algorithm to use at least (Formula presented) different colors on an (Formula presented) -colorable set of unit intervals.