Data Structures for Incremental Interval Coloring

TLDR: This work designs an incremental algorithm that is subtly different from the KT-algorithm and uses at most \(3 \omega - 2\) colors, where \(\omega \) is the maximum clique in the interval graph associated with the set of intervals.

Abstract: We consider the dynamic graph coloring problem restricted to the class of interval graphs. At each update step the algorithm is presented with an interval to be colored, or a previously colored interval to delete. The goal of the algorithm is to efficiently maintain a proper coloring of the intervals with as few colors as possible by an online algorithm. In the incremental model, each update step presents the algorithm with an interval to be colored. The problem is closely connected to the online vertex coloring problem of interval graphs for which the Kierstead-Trotter (KT) algorithm achieves the best possible competitive ratio. We first show that a sub-quadratic time direct implementation of the KT-algorithm is unlikely to exist conditioned on the correctness of the Online Boolean Matrix Vector multiplication conjecture due to Henzinger et al. [9]. We then design an incremental algorithm that is subtly different from the KT-algorithm and uses at most \(3 \omega - 2\) colors, where \(\omega \) is the maximum clique in the interval graph associated with the set of intervals. Our incremental data structure maintains a proper coloring in amortized \(O(\log n + \varDelta )\) update time where n is the total number of intervals inserted and \(\varDelta \) is the maximum degree of a vertex in the interval graph.