A New Perspective on the Windows Scheduling Problem

Abstract

The Windows Scheduling Problem, also known as the Pinwheel Problem, is to schedule periodic jobs subject to their processing frequency demands. Instances are given as a set of jobs that have to be processed infinitely often such that the time interval between two consecutive executions of the same job j is no longer than the job's given period pj .The key contribution of this work is a new interpretation of the problem variant with exact periods, where the time interval between consecutive executions must be strictly pj. We show that this version is equivalent to a natural combinatorial problem we call Partial Coding. Reductions in both directions can be realized in polynomial time, so that both hardness proofs and algorithms for Partial Coding transfer to Windows Scheduling.Applying this new perspective, we obtain a number of new results regarding the computational complexity of various Windows Scheduling Problem variants. We prove that even the case of one processor and unitlength jobs does not admit a pseudo-polynomial time algorithm unless SAT can be solved by a randomized method in expected quasi-polynomial time. This result also extends to the case of inexact periods, which answers a question that has remained open for more than two decades. Furthermore, we report an error found in a hardness proof previously given for the multi-machine case without machine migration, and we show that this variant reduces to the single-machine case. Finally, we prove that even with unit-length jobs the problem is co-NP-hard when jobs are allowed to migrate between machines.

Blocking Bandits

“…Computational Hardness: We prove that when the rewards and the delays are known, the problem of choosing a sequence of available arms to optimize the reward over a time horizon T is computationally hard (see, Theorem 3.1). Specifically, we prove the offline optimization is as hard as PINWHEEL Scheduling on dense instances [17,11,18,3], which does not permit any pseudo-polynomial time algorithm (in the number of arms) unless randomized exponential time hypothesis [5] is false.…”

“…Proof. The proof follows from Theorem 3.1 and Theorem 24 in [18]. In [18], the authors shows that the PINWHEEL SCHEDULING with dense instances do not admit any pseudo-polynomial algorithm unless the randomized exponential time hypothesis [5] is False.…”

“…A PINWHEEL SCHEDULING instance is called dense if K i=1 1/a i = 1. Also, note that this problem is also known as Single Machine Windows Scheduling Problem with Inexact Periods [18]. Theorem 3.1.…”

Recurrent Submodular Welfare and Matroid Blocking Bandits

“…decide whether there is a coloring of the natural numbers ν : N → [k] such that every color i ∈ [k] appears at least once every d i numbers. As it is proved in [23], the above problem does not admit a pseudopolynomial time algorithm unless SAT can be solved by a randomized algorithm in expected quasi-polynomial time.…”

Periodic Scheduling and Packing Problems

“…Periodic scheduling on one machine with unit processing times was shown to be NP-complete by [1] using the reduction from the graph coloring problem. Furthermore, [6] proved that the problem is strongly NP-hard.…”