Sublinear Dynamic Interval Scheduling (on one or multiple machines)

We revisit the complexity of the classical Interval Scheduling in the dynamic setting. In this problem, the goal is to maintain a set of intervals under insertions and deletions and report the size of the maximum size subset of pairwise disjoint intervals after each update. Nontrivial approximation algorithms are known for this problem, for both the unweighted and weighted versions [Henzinger, Neumann, Wiese, SoCG 2020]. Surprisingly, it was not known if the general exact version admits an exact solution working in sublinear time, that is, without recomputing the answer after each update. Our first contribution is a structure for Dynamic Interval Scheduling with amortized $\tilde{\mathcal{O}}(n^{1/3})$ update time. Then, building on the ideas used for the case of one machine, we design a sublinear solution for any constant number of machines: we describe a structure for Dynamic Interval Scheduling on $m\geq 2$ machines with amortized $\tilde{\mathcal{O}}(n^{1 - 1/m})$ update time. We complement the above results by considering Dynamic Weighted Interval Scheduling on one machine, that is maintaining (the weight of) the maximum weight subset of pairwise disjoint intervals. We show an almost linear lower bound (conditioned on the hardness of Minimum Weight $k$-Clique) for the update/query time of any structure for this problem. Hence, in the weighted case one should indeed seek approximate solutions.