Truthfulness for the Sum of Weighted Completion Times

Abstract

We consider the problem of designing truthful mechanisms for scheduling selfish tasks on a single machine or on a set of m parallel machines. The objective of every selfish task is the minimization of its completion time while the aim of the mechanism is the minimization of the sum of weighted completion times. For the model without payments, we prove that there is no $(2-\epsilon )$-approximate deterministic truthful algorithm and no $(\frac{3}{2}-\epsilon )$-approximate randomized truthful algorithm when the tasks’ lengths are private data. When both the lengths and the weights are private data, we show that it is not possible to get an $\alpha$-approximate deterministic truthful algorithm for any $\alpha >1$. In order to overcome these negative results we introduce a new concept that we call preventive preemption. Using this concept, we are able to propose a simple optimal truthful algorithm with no payments for the single-machine problem when the lengths of the tasks are private. For multiple machines, we present an optimal truthful algorithm for the unweighted case. For the weighted-multiple-machines case, we propose a truthful randomized algorithm which is $\frac{3}{2}$-approximate in expectation based on preventive preemption. For the model with payments, we prove that there is no optimal truthful algorithm even when only the lengths of the tasks are private data. Then, we propose an optimal truthful mechanism using preventive preemption and appropriately chosen payments.