Maximizing the minimum load for selfish agents

We consider the problem of maximizing the minimum load (completion time) for machines that are controlled by selfish agents, who are only interested in maximizing their own profit. Unlike the classical load balancing problem, this problem has not been considered for selfish agents until now. The goal is to design a truthful mechanism, i.e., one in which all users have an incentive to tell the truth about the speeds of their machines. This then allows us to find good job assignments. It is known that this requires monotone approximation algorithms, in which the amount of work assigned to an agent does not increase if its bid (claimed cost per unit work) increases. For a constant number of machines, m, we show a monotone polynomial-time approximation scheme (PTAS) with running time that is linear in the number of jobs. It uses a new technique for reducing the number of jobs while remaining close to the optimal solution. We use an FPTAS for the classical problem, i.e., where no selfish agents are involved, to give a monotone FPTAS. Additionally, we give a monotone approximation algorithm with approximation ratio min (m, (2 + ε) s₁ / s_m) where ε > 0 can be chosen arbitrarily small and s_i is the (real) speed of machine i. Finally we give improved results for two machines.