Scheduling in the Secretary Model

TL;DR

This paper investigates online makespan minimization in the secretary model, presenting new algorithms with improved competitive ratios that perform well on most job orderings, contrasting with worst-case scenarios.

Contribution

It introduces two deterministic algorithms with competitive ratios of 1.75 and 1.535, and provides lower bounds, highlighting the effectiveness of randomization in the secretary model.

Findings

Deterministic algorithm with 1.75 competitive ratio

New algorithm achieving 1.535 competitive ratio

Lower bounds indicating limits of online algorithms

Abstract

This paper studies Makespan Minimization in the secretary model. Formally, jobs, specified by their processing times, are presented in a uniformly random order. An online algorithm has to assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized. We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy LightLoad provides a very simple algorithm retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders. Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications. They also demonstrate the power of randomization when compared to much stronger deterministic reordering models. We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model.