Maximizing the Minimum Load for Selfish Agents

We consider the problem of maximizing the minimum load for machines that are controlled by selfish agents, who are only interested in maximizing their own profit. Unlike the regular load balancing problem, this problem has not been considered in this context before. 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 also present an FPTAS for the classical machine covering problem, i.e., where no selfish agents are involved (the previous best result for this case was a PTAS) and use this 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.