We consider the problem of designing truthful mechanisms for scheduling n tasks on a set of m parallel related machines in order to minimize the makespan. In what follows, we consider that each task is owned by a selfish agent. This is a variant of the KP-model introduced by Koutsoupias and Papadimitriou (1999) (and of the CKN-model of Christodoulou et al. (2004)) in which the agents cannot choose the machine on which their tasks will be executed. This is done by a centralized authority, the scheduler. However, the agents may manipulate the scheduler by providing false information regarding the length of their tasks. We introduce the notion of increasing algorithm and a simple reduction that transforms any increasing algorithm into a truthful one. Furthermore, we show that some of the classical scheduling algorithms are indeed increasing: the Longest Processing Time (LPT) algorithm, the Polynomial-Time Approximation Scheme(PTAS) of Graham (1969) in the case of two machines, as well as a simple PTAS for the case of m machines, with m a fixed constant. Our results yield a randomized r(1+ϵ)-approximation algorithm where r is the ratio between the largest and the smallest speed of the related machines. Furthermore, by combining our approach with the classical result of Shmoys et al. (1995), we obtain a randomized 2r(1+ϵ)-competitive algorithm. It has to be noticed that these results are obtained without payments, unlike most of the existing works in the field of Mechanism Design. Finally, we show that if payments are allowed then our approach gives a (1+ϵ)-algorithm for the off-line case with related machines.
