An Approximation Algorithm for Scheduling Deteriorating Jobs

In scheduling theory, the non-preemptive scheduling on a single machine of jobs with increasing processing times and release dates for total completion time minimization is known to be a strongly NP-hard problem. Consequently, we cannot construct an exact polynomial-time algorithm for this problem, prompting the exploration of approximation algorithms as the best viable approach. Nothing was known in the literature about constructing good approximations in polynomial time for this complex problem. Our approach to constructing an approximation algorithm involves two key steps. First, we develop an optimal algorithm for the preemptive version of the scheduling problem. Building upon this foundation, we then develop the core of our approximation algorithm. However, due to the intricate nature of the problem, deriving a precise performance ratio for the proposed approximation algorithm is impossible. The performance ratio varies depending on the problem instance. Nevertheless, we provide an instance to demonstrate that the approximation ratio could be at least 2.