Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints

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 (5m - 3) log d/log d+3 different colors on an m(d/k + log d + 3)-colorable set of intervals. The second strategy forces any on-line interval coloring algorithm to use at least [5m/2] d/log d+3 different colors on an m(d/k + log d + 3)-colorable set of unit intervals.