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.

The longest common substring problem consists in finding a longest string that appears as a (contiguous) substring of two input strings. We consider the dynamic variant of this problem, in which we are to maintain two dynamic strings S and T, each of length at most n, that undergo substitutions of letters, in order to be able to return a longest common substring after each substitution. Recently, Amir et al. [ESA 2019] presented a solution for this problem that needs only 𝒪̃(n^(2/3)) time per update. This brought the challenge of determining whether there exists a faster solution with polylogarithmic update time, or (as is the case for other dynamic problems), we should expect a polynomial (conditional) lower bound. We answer this question by designing a significantly faster algorithm that processes each substitution in amortized log^𝒪(1) n time with high probability. Our solution relies on exploiting the local consistency of the parsing of a collection of dynamic strings due to Gawrychowski et al. [SODA 2018], and on maintaining two dynamic trees with labeled bicolored leaves, so that after each update we can report a pair of nodes, one from each tree, of maximum combined weight, which have at least one common leaf-descendant of each color. We complement this with a lower bound of Ω(log n/ log log n) for the update time of any polynomial-size data structure that maintains the LCS of two dynamic strings, even allowing amortization and randomization.