Scheduling with Obligatory Tests

TL;DR

This paper studies a scheduling problem involving jobs with tests and processing, providing algorithms with competitive ratios and establishing lower bounds for deterministic algorithms in such settings.

Contribution

The paper introduces a novel analysis technique and demonstrates that the 1-SORT algorithm achieves a competitive ratio of at most 1.861, with improved ratios for special cases.

Findings

1-SORT algorithm has a competitive ratio at most 1.861.

A threshold-based algorithm achieves a ratio at most 1.585 for uniform test times.

No deterministic algorithm can have a competitive ratio better than √2.

Abstract

Motivated by settings such as medical treatments or aircraft maintenance, we consider a scheduling problem with jobs that consist of two operations, a test and a processing part. The time required to execute the test is known in advance while the time required to execute the processing part becomes known only upon completion of the test. We use competitive analysis to study algorithms for minimizing the sum of completion times for $n$ given jobs on a single machine. As our main result, we prove using a novel analysis technique that the natural $1$-SORT algorithm has competitive ratio at most 1.861. For the special case of uniform test times, we show that a simple threshold-based algorithm has competitive ratio at most 1.585. We also prove a lower bound that shows that no deterministic algorithm can be better than $\sqrt{2}$-competitive even in the case of uniform test times.