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 ${\left( 5m-3\right) }\frac{d}{\log d + 3}$ different colors on an $m{\left( \frac{d}{k} + \log {d} + 3\right) }$-colorable set of intervals. The second strategy forces any on-line interval coloring algorithm to use at least ${\left\lfloor \frac{5m}{2} \right\rfloor }\frac{d}{\log d + 3}$ different colors on an $m{\left( \frac{d}{k} + \log {d} + 3\right) }$-colorable set of unit intervals.

Research partially supported by NCN grant number 2014/14/A/ST6/00138.