Online Capacitated Interval Coloring

In the online capacitated interval coloring problem, a sequence of requests arrive online. Each of the requests is an interval Ij ⊆ {1, 2, . . . , n} with bandwidth bj . Initially a vector of capacities (c1, c2, . . . , cn) is given. Each color can support a set of requests such that the total bandwidth of intervals containing i is at most ci. 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 bmax = maxj bj is at most the minimum capacity cmin = mini ci. For the case bmax > cmin, we give an algorithm with competitive ratio O(log bmax cmin ) and, using resource augmentation, a constant competitive algorithm. We also give a lower bound showing that constant competitive ratio cannot be achieved in this case without resource augmentation.