Randomized Truthful Mechanisms for Scheduling Unrelated Machines

In this paper, we consider randomized truthful mechanisms forscheduling tasks to unrelated machines, where each machine iscontrolled by a selfish agent. Some previous work on this topicfocused on a special case, scheduling two machines, for which thebest approximation ratio is 1.6737 [5] and the best lower bound is1.5 [6]. For this case, we give a unified framework for designinguniversally truthful mechanisms, which includes all the knownmechanisms, and also a tight analysis method of their approximationratios. Based on this, we give an improved randomized truthfulmechanism, whose approximation ratio is 1.5963. For the generalcase, when there are m machines, the only known technique is toobtain a $\frac {\gamma m}{2}$-approximation truthful mechanism bygeneralizing a γ-approximation truthful mechanism for twomachines[6]. There is a barrier of 0.75m for this technique due tothe lower bound of 1.5 for two machines. We break this 0.75mbarrier by a new designing technique, rounding a fractionalsolution. We propose a randomized truthful-in-expectation mechanismthat achieves approximation of $\frac{m+5}{2}$, for m machines. For the lower bound side, we focus on an interesting family ofmechanisms, namely task-independent truthful mechanisms.We prove a lower bound of 11/7 for two machines and a lower boundof $\frac{m+1}{2}$ for m machines with respect to thisfamily. They almost match our upper bounds in both cases.