Online capacitated interval coloring

In the online capacitated interval coloring problem, a sequence of requests arrive online. Each request is an interval I_j ⊆ {1, 2, ⋯, n} with bandwidth b_j. We are initially given a vector of capacities (c₁, c₂, ⋯, 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_max = max_j b_j is at most the minimum capacity c_min = min_i c _i. For the case b_max > c_min, we give an algorithm with competitive ratio O(log b_max/c_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.