Online Capacitated Interval Coloring

In the online capacitated interval coloring problem, a sequence of requests arrive online. Each request is an interval $I_j\subseteq\{1,2,\dots,n\}$ with bandwidth $b_j$. We are initially given a vector of capacities $(c_1,c_2,\dots,c_n)$. Each color can support a set of requests such that the total bandwidth of intervals containing $i$ is at most $c_i$. The goal is to color the requests using a minimum number of colors. We present a constant competitive algorithm for the case where the maximum bandwidth $b_{\mathrm{max}}=\max_j b_j$ is at most the minimum capacity $c_{\mathrm{min}}=\min_i c_i$. For the case $b_{\mathrm{max}}>c_{\mathrm{min}}$, we give an algorithm with competitive ratio $O(\log\frac{b_{\mathrm{max}}}{c_{\mathrm{min}}})$ and, using resource augmentation, a constant competitive algorithm. We also give a lower bound showing that a constant competitive ratio cannot be achieved in the general case without resource augmentation.