Multi-Organizational Scheduling: Individual Rationality, Optimality, and Complexity

We investigate multi-organizational scheduling problems, building upon the framework introduced by Pascual et al.[2009]. In this setting, multiple organizations each own a set of identical machines and sequential jobs with distinct processing times. The challenge lies in optimally assigning jobs across organizations' machines to minimize the overall makespan while ensuring no organization's performance deteriorates. To formalize this fairness constraint, we introduce individual rationality, a game-theoretic concept that guarantees each organization benefits from participation. Our analysis reveals that finding an individually rational schedule with minimum makespan is $\Theta_2^{\text{P}}$-hard, placing it in a complexity class strictly harder than both NP and coNP. We further extend the model by considering an alternative objective: minimizing the sum of job completion times, both within individual organizations and across the entire system. The corresponding decision variant proves to be NP-complete. Through comprehensive parameterized complexity analysis of both problems, we provide new insights into these computationally challenging multi-organizational scheduling scenarios.