The Offline Carpool Problem Revisited

Abstract

The carpool problem is to schedule for every time $t\in \mathbb {N}$ l tasks taken from the set [n] ($n\ge 2$). Each task i has a weight $w_{i}(t)\ge 0$, where $\sum _{i=1}^n w_{i}(t)=l$. We let $c_i(t)\in \{0,1\}$ be 1 iff task i is scheduled at time t, where (carpool condition) $w_i(t)=0\Rightarrow c_i(t)=0$.

The carpool problem exists in the literature for $l=1$, with a goal to make the schedule fair, by bounding the absolute value of $E_i(t)=\sum _{s=1}^t[w_{i}(s)-c_{i}(s)]$. In the typical online setting, $w_i(t)$ is unknown prior to time t; therefore, the only sensible approach is to bound $|E_i(t)|$ at all times. The optimal online algorithm for $l=1$ can guarantee $|E_i(t)|=O(n)$. We show that the same guarantee can be maintained for a general l. However, it remains far from an ideal $|E_i(T)|<1$ 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 $w_i(t)$ is known in advance for all times $t\le T$, and the fairness requirement is strengthened to the ideal $|E_i(T)|<1$ while keeping $E_i(t)$ 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 $l\ge 1$. We show that achieving the ideal fairness with an intermediate $O(n^2)$ bound is possible for a general l.

S. Mneimneh—Partially supported by the CoSSMO institute at CUNY.

S. Farhat—Supported by a CUNY Graduate Center Fellowship.