Scheduling with a Limited Testing Budget

TL;DR

This paper studies scheduling problems with limited testing budgets, providing approximation algorithms and complexity results for both offline and oblivious variants under total completion time and makespan objectives.

Contribution

It extends previous work by incorporating testing costs and budgets, offering new PTAS, FPTAS, and competitive algorithms for these scheduling problems.

Findings

NP-hardness for total completion time with testing budget

PTAS for offline total completion time scheduling

$(4+ ext{epsilon})$-competitive algorithm for oblivious total completion time scheduling

Abstract

Scheduling with testing falls under the umbrella of the research on optimization with explorable uncertainty. In this model, each job has an upper limit on its processing time that can be decreased to a lower limit (possibly unknown) by some preliminary action (testing). Recently, D{\"{u}}rr et al. \cite{DBLP:journals/algorithmica/DurrEMM20} has studied a setting where testing a job takes a unit time, and the goal is to minimize total completion time or makespan on a single machine. In this paper, we extend their problem to the budget setting in which each test consumes a job-specific cost, and we require that the total testing cost cannot exceed a given budget. We consider the offline variant (the lower processing time is known) and the oblivious variant (the lower processing time is unknown) and aim to minimize the total completion time or makespan on a single machine. For the total completion time objective, we show NP-hardness and derive a PTAS for the offline variant based on a novel LP rounding scheme. We give a $(4+\epsilon)$-competitive algorithm for the oblivious variant based on a framework inspired by the worst-case lower-bound instance. For the makespan objective, we give an FPTAS for the offline variant and a $(2+\epsilon)$-competitive algorithm for the oblivious variant. Our algorithms for the oblivious variants under both objectives run in time $O(poly(n/\epsilon))$. Lastly, we show that our results are essentially optimal by providing matching lower bounds.