Single-machine scheduling with multi-agents to minimize total weighted late work

We consider the competitive multi-agent scheduling problem on a single machine, where each agent's cost function is to minimize its total weighted late work. The aim is to find the Pareto-optimal frontier, i.e., the set of all Pareto-optimal points. When the number of agents is arbitrary, the decision problem is shown to be unary NP-complete even if all jobs have the unit weights. When the number of agents is two, the decision problems are shown to be binary NP-complete for the case in which all jobs have the common due date and the case in which all jobs have the unit processing times. When the number of agents is a fixed constant, a pseudo-polynomial dynamic programming algorithm and a (1+epsilon)-approximate Pareto-optimal frontier are designed to solve it.