A note on the online interval scheduling secretary problem

Abstract

The interval scheduling secretary problem models the scenario when a number of job intervals arrive randomly and the decider needs to assign some of them to machines irrevocably such that all assigned job intervals can be processed by machines without overlap, with the goal of maximizing the social welfare. Im and Wang ((2011) [7]) designed an almost optimal logarithm-competitive online algorithm for the single-machine case. In this note, we extend this result to multiple heterogeneous machines. Our algorithm preserves the competitive ratio and is thus nearly optimal.

Introduction

In this note, we extend the interval scheduling secretary problem (ISSP) initiated by [7] to the setting of multiple heterogeneous machines. In an ISSP, a number of job intervals, with release time, deadline, processing time and value (possibly different to different machines), arrive in a random order. After each job arrives, a decision must be made immediately and irrevocably: assign it to one of the machines, or reject it. It is required that all assigned jobs to one machine should be able to be scheduled, i.e., there exists a schedule such that every assigned job can be completed within its release time and deadline. Completing each job brings the corresponding value, and the objective is to make a schedulable assignment so as to maximize the expectation of the total value.

Our research is partly motivated by the following problem faced by the Students Affairs Office, who assigns randomly arriving part-time jobs to the students that submitted job applications. Each arrived part-time job requires a fixed time length to complete, but the exact process duration may have some flexibility. For example, an one-hour peer coaching needs to be scheduled on September 28th, and any consecutive one hour between 2:00 pm and 5:00 pm works well. To provide timely customer service, each job needs to be assigned to one of the applicants as soon as it arrives or declined to be scheduled immediately. A feasible schedule requires that the jobs assigned to an applicant can be completed without mutual overlap. The students may hold different job preferences, and the preferences are revealed upon the arrival of each job. We want to understand the extent to which the office can maximize the expected social welfare. There are other real-world scenarios that can be captured by our model, like Uber ride-scheduling and food-delivery [5], and edge-cloud computing problems [12], where the requests (i.e., ride/food orders or jobs released from mobile devices/users) can be abstracted by time intervals and as soon as a request arrives, the system assigns it to a driver or an edge server.

The ISSP is essentially a generalization of the classical secretary problem, which has been widely concerned in both artificial intelligence [1], [2], [6] and theoretical computer science [3], [4], [7] due to its practical and theoretical importance. Some other variants of the secretary problem, include selecting up to a fixed number of items [9], the selected items should satisfy a matroid [4] or knapsack constraint [3], [11], and selecting items that form a matching in a graph [8], [10].

For the ISSP with a single machine, in which the decider does not need to assign the jobs but just accepts or rejects each job, Im and Wang [7] proposed a framework and designed an $O(\log D)$-competitive algorithm, where D is the maximum length of any job interval. They also proved that this algorithm is almost optimal by giving a lower bound $\mathrm{\Omega }(\frac{\log D}{\log \log D})$. Preemption is allowed in their algorithm, which informally means that a job is not necessarily to be processed continuously. In this note, we extend their result to more general settings with multiple heterogeneous machines (i.e., a job may bring different values to different machines). Our algorithm preserves the competitive ratio $O(\log D)$ and is thus almost optimal. Moreover, our result continues to hold even if preemption is not allowed. Our technique is grounded on the framework in [7] and utilizes the recent progresses in other variants of secretary problem [8], [11].