New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results. For $(1+\varepsilon)$-approximation of job scheduling of $n$ jobs on a single machine, we develop a fully dynamic algorithm with $O(\frac{\log{n}}{\varepsilon})$ update and $O(\log{n})$ query worst-case time. Further, we design a local computation algorithm that uses only $O(\frac{\log{N}}{\varepsilon})$ queries when all jobs are length at least $1$ and have starting/ending times within $[0,N]$. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we design a fully dynamic deterministic algorithm whose worst-case update and query time are $\operatorname{poly}(\log n,\frac{1}{\varepsilon})$. Equivalently, this is the first algorithm that maintains a $(1+\varepsilon)$-approximation of the maximum independent set of a collection of weighted intervals in $\operatorname{poly}(\log n,\frac{1}{\varepsilon})$ time updates/queries. This is an exponential improvement in $1/\varepsilon$ over the running time of a randomized algorithm of Henzinger, Neumann, and Wiese ~[SoCG, 2020], while also removing all dependence on the values of the jobs' starting/ending times and rewards, as well as removing the need for any randomness. We also extend our approaches for interval scheduling on a single machine to examine the setting with $M$ machines.