New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Abstract

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 + ε)-approximation of job scheduling of n jobs on a single machine, we obtain a fully dynamic algorithm with O( log n /ε) update and O(log n) query worst-case time. Further, we design a local computation algorithm that uses only O( log n /ε) queries. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we obtain a fully dynamic algorithm whose worst-case update and query time has only polynomial dependence on 1/ε, which is an exponential improvement over the result of Henzinger et al. [SoCG, 2020].We extend our approaches for unweighted interval scheduling on a single machine to the setting with M machines, while achieving the same approximation factor and only M times slower update time in the dynamic setting. In addition, we provide a general framework for reducing the task of interval scheduling on M machines to that of interval scheduling on a single machine. In the unweighted case this approach incurs a multiplicative approximation factor 2 − 1/M .

Sublinear Dynamic Interval Scheduling (on one or multiple machines)

“…For the general version of DIS, Henzinger, Neumann and Wiese [15] designed an efficient approximation algorithm that maintains an (1 + )-approximate solution in polylogarithmic time. The dependency on has been very recently improved from exponential to polynomial by Compton, Mitrović and Rubinfeld [7]. In fact, both solutions work for the weighted version of the problem, called Dynamic Weighted Interval Scheduling (DWIS).…”

“…The weighted version of the problem (WIS+) can be formulated and solved as a min-cost flow problem [3,5]. For the dynamic version, Compton, Mitrović and Rubinfeld [7] extend their methods for maintaining an approximate answer to multiple machines, however, their bounds are mostly relevant for the unweighted case. A related (but not directly connected) question is to maintain the smallest number of machines necessary to schedule all jobs in the current set [12].…”

On Streaming Algorithms for Geometric Independent Set and Clique

“…Very recently, this factor was improved to 3 [23]. Also, the geometric maximum independent set problem has been extensively studied for dynamic geometric objects, i.e., objects can be inserted and deleted [9,11,16,25,28].…”