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

TL;DR

This paper establishes lower bounds for online interval coloring algorithms under vector and cardinality resource constraints, highlighting the difficulty of resource-aware scheduling in online settings.

Contribution

It introduces two strategies that demonstrate the minimum number of colors any online algorithm must use under specific resource constraints, extending previous models.

Findings

Lower bounds on color usage for online algorithms with vector constraints

Lower bounds for algorithms with cardinality constraints

Implications for resource-aware task scheduling

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