Dynamic data structures for interval coloring

We consider the dynamic graph coloring problem restricted to the class of interval graphs in the incremental and fully dynamic setting. The input consists of a sequence of intervals that are to be either colored, or deleted, if previously colored. For the incremental setting, we consider the well studied optimal online algorithm (KT-algorithm) for interval coloring due to Kierstead and Trotter [1]. We present the following results on the dynamic interval coloring problem. ■ Any direct implementation of the KT-algorithm requires Ω(Δ2) time per interval in the worst case. ■ There exists an incremental algorithm which supports insertion of an interval in amortized O(log⁡n+Δ) time per update and maintains a proper coloring using at most 3ω−2 colors. ■ There exists a fully dynamic algorithm which supports insertion of an interval in O(log⁡n+Δlog⁡ω) update time and deletion of an interval in O(Δ2log⁡n) update time in the worst case and maintains a proper coloring using at most 3ω−2 colors. The KT-algorithm crucially uses the maximum clique size in an induced subgraph in the neighborhood of a given vertex. We show that the problem of computing the induced subgraph among the neighbors of a given vertex has the same hardness as the online boolean matrix vector multiplication problem [2]. We show that ■ Any algorithm that computes the induced subgraph among the neighbors of a given vertex requires at least quadratic time unless the OMv conjecture [2] is false. Finally, we obtain the following result on the OMv conjecture. ■ If the matrix and the vectors in the online sequence have the consecutive ones property, then the OMv conjecture [2] is false.