Extremal behavior of the Greedy algorithm for a triangle scheduling problem

Abstract

We study the mixed-criticality scheduling problem, where the goal is to schedule jobs with different criticality levels on a single machine. As shown by Dürr et al. (2018), the problem can be treated as a specific 1-dimensional triangle scheduling problem. In that paper a new Greedy algorithm was defined, and the authors proved that its approximation ratio lies between 1.05 and 3/2. In this paper we present a quadratic integer programming model, which can be used to computationally analyze the algorithm for inputs with small sizes. The model simulates the behavior of the algorithm and it compares the makespan with the optimal one. Using this model, we found sequences extendable to longer series, giving a lower bound of 1.27 for the Greedy algorithm. Also, the optimum on problem instances consisting of intervals of natural numbers is analyzed and a closed formula is determined. In this way, we detected two input classes where, in one of them, Greedy is far from optimal (we think that this could be the worst case), and in the other one it is optimal.