Competitive Algorithms for the Online Minimum Peak Appointment Scheduling

Abstract

This paper describes a fundamental online scheduling problem called the minimum peak appointment scheduling (MPAS) problem. In this problem, there is a sequence of arriving appointments, each with specified required duration. The goal is to schedule the appointments in an online manner and to minimize the resulting peak utilization throughout the scheduling interval. While described as appointment scheduling, the MPAS has far broader applicability that captures other settings. The paper provides a formal modeling framework to analyze the aforementioned MPAS problem. It is shown that the offline version of the MPAS problem is identical to the offline version of the well-known bin packing problem. However, in the online variant, there is additional flexibility regarding how appointments are packed into chairs (bins). The paper describes the first competitive online algorithm to the MPAS problem called the harmonic rematching (HR) algorithm. The analysis shows that the HR algorithm has an asymptotic competitive ratio of 1.5. Considering that the current best lower bound on randomized online algorithms is 1.536, this highlights the fact that the HR algorithm performs better than any bin packing online algorithm in this setting.