Single-Machine Scheduling Problems with Two Agents Competing for Makespan

This paper considers the single-machine scheduling problems in which two distinct agents are concerned. Each agent has a set of jobs with different release times, and both of them expect to complete their respective jobs as early as possible. We take the makespan of each agent as its own criterion and take the linear combination of the two makespans as our objective function. In this paper, both off-line and on-line models are considered. When preemption is allowed, we present an exact algorithm for the off-line model and an optimal algorithm for the on-line model. When preemption is not allowed we point out that the problem is NP-hard for the off-line model and give a (2+1/theta)-competitive algorithm for the on-line model. We also prove that a lower bound of the competitive ratio for the later model is 1 + theta/(1 + theta), where theta is a given factor not less than 1.