New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Interval scheduling is a basic algorithmic problem and a classical task in combinatorial optimization. We develop techniques for partitioning and grouping jobs based on their starting/ending times, enabling 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 a dynamic setting produces several new results. For ( 1 + ε ) -approximation of job scheduling of n jobs on a single machine, we develop a fully dynamic algorithm with O ( log n ε ) update and O ( log n ) query worst-case time. Our techniques are also applicable in a setting where jobs have weights. We design a fully dynamic deterministic algorithm whose worst-case update and query times are poly ( log n , 1 ε ) . This is the first algorithm that maintains a ( 1 + ε ) -approximation of the maximum independent set of a collection of weighted intervals in poly ( log n , 1 ε ) time updates/queries. This is an exponential improvement in 1 / ε over the running time of an algorithm of Henzinger, Neumann, and Wiese [SoCG, 2020]. Our approach also removes all dependence on the values of the jobs’ starting/ending times and weights.