The offline carpool problem revisited

Abstract

The carpool problem is to schedule for every time t ∈ ℕ l tasks taken from the set [n] (n ≥ 2). Each task i has a weight wi(t) ≥ 0, where ∑ni=1 wi(t) = l. We let ci(t) ∈ {0, 1} be 1 iff task i is scheduled at time t, where (carpool condition) wi(t) = 0 ⇒ ci(t) = 0. The carpool problem exists in the literature for l = 1, with a goal to make the schedule fair, by bounding the absolute value of Ei(t) = ∑ts=1[wi(s) − ci(s)]. In the typical online setting, wi(t) is unknown prior to time t; therefore, the only sensible approach is to bound |Ei(t)| at all times. The optimal online algorithm for l = 1 can guarantee |Ei(t)| = O(n). We show that the same guarantee can be maintained for a general l. However, it remains far from an ideal |Ei(T)| < 1 when all tasks have reached completion at some future time t = T. The main contribution of this paper is the offline version of the carpool problem, where wi(t) is known in advance for all times t ≤ T, and the fairness requirement is strengthened to the ideal |Ei(T)| < 1 while keeping Ei(t) bounded at all intermediate times t < T. This problem has been mistakenly considered solved for l = 1 using Tijdeman’s algorithm, so it remains open for l ≥ 1. We show that achieving the ideal fairness with an intermediate O(n2) bound is possible for a general l.