Bounds on the average and minimum attendance in preference-based activity scheduling

Abstract

In this paper we study the preference-based activity scheduling problem, where based on preference sets of participants, activities need to be scheduled in parallel, such that participants can attend as many of their preferred activities as possible. The Activity Scheduling Problem (ASP) takes as input the activities, time slots, participants and their set of preferences, and is supposed to output an optimal schedule. Attendance is the number of preferred activities a participant can attend. We present two lower bounds on the average attendance over all participants, based on linearity of expectation. The first is a bound that involves the different parameters of ASP; we prove that this bound is tight. The second bound is such that no matter how many activities, participants, and time slots are included in the event, there is always a schedule that guarantees an average attendance of (1−1/e) of the preferred activities for the participants. A further result, based on applying the Lovász local lemma to hypergraph coloring, presents a lower bound on the minimum attendance over all participants. We show that for various values of the problem’s parameters this bound is better than a simple bound, although the simple bound may be tight for instances with a very large number of participants.