Result: Truthful algorithms for scheduling selfish tasks on parallel machines

We consider the problem of designing truthful mechanisms for scheduling selfish tasks (or agents)-whose objective is the minimization of their completion times¯on parallel identical machines in order to minimize the makespan. A truthful mechanism can be easily obtained in this context (if we, of course, assume that the tasks cannot shrink their lengths) by scheduling the tasks following the increasing order of their lengths. The quality of a mechanism is measured by its approximation factor (price of anarchy, in a distributed system) w.r.t. the social optimum. The previous mechanism, known as SPT, produces a (2 - 1/m)-approximate schedule, where m is the number of machines. The central question in this paper is the following: Are there other truthful mechanisms with better approximation guarantee (price of anarchy) for the considered scheduling problem? This question has been raised by Christodoulou et al. [Coordination mechanisms, in: Proc. of ICALP 2004, Lecture Notes in Computer Science, Vol. 3142, 345-357.] in the context of coordination mechanisms, but it is also relevant in centrally controlled systems. We present (randomized) truthful mechanisms for both the centralized and the distributed settings that improve the (expected) approximation guarantee (price of anarchy) of the SPT mechanism. Our centralized mechanism holds for any number of machines and arbitrary task lengths, while the coordination mechanism holds only for two machines and task lengths that are powers of a certain constant.