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.
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References
Fagin, R., Williams, J.H.: A fair carpool scheduling algorithm. IBM J. Res. Dev. 27(2), 133–139 (1983)
Mneimneh, S.: Load balancing in a switch without buffers. In: IEEE Workshop on High Performance Switching and Routing, Poznan (2006)
Coppersmith, D., Nowicki, T., Paleologo, G., Tresser, C., Wu, C.W.: The optimality of the online greedy algorithm in carpool and chairman assignment problems. ACM Trans. Algorithms, 7(3), Article 37, July 2011
Ajtai, M., Aspnes, J., Naor, M., Rabini, Y., Schulman, L.J., Waarts, O.: Fairness in Scheduling. J. Algorithms 29(2), 306–357 (1988)
Naor, M.: How to Carpool Fairly. http://www.wisdom.weizmann.ac.il/naor/PAPERS/carpool_fair.pps
Naor, M.: On fairness in the carpool problem. J. Algorithms 55(1), 93–98 (2005)
Williamson, D.: Lecture Notes on Network Flows, Chapter 3. http://people.orie.cornell.edu/dpw/techreports/cornell-flow.pdf
Havet, F.: Combinatorial Optimization, Chapter 11 on Fractional Relaxation. http://www-sop.inria.fr/members/Frederic.Havet/
Boavida, J.B., Kamat, V., Nakum, D., Nong, R., Wu, C.W., Zhang, X.: Algorithms for the Carpool Problem. http://www.ima.umn.edu/2005-2006/MM8.9-18.06/activities/Wu-Chai/team6_rep.pdf
Tijdeman, R.: The chairman assignment problem. Discrete Math. 32, 323–330 (1980)
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Mneimneh, S., Farhat, S. (2015). The Offline Carpool Problem Revisited. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_40
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