On the Structure of Optimal Greedy Computation (for Job Scheduling)

We consider Priority Algorithm [4] as a syntactic model of formulating the concept of greedy algorithm for Job Scheduling, and we study the computation of optimal priority algorithms. A Job Scheduling subproblem $\mathbb{S}$ is determined by a (possibly infinite) set of jobs, every finite subset of which potentially forms an input to a scheduling algorithm. An algorithm is optimal for $\mathbb{S}$, if it gains optimal profit on every input. To the best of our knowledge there is no previous work about such arbitrary subproblems of Job Scheduling. For a finite $\mathbb{S}$, it is coNP-hard to decide whether $\mathbb{S}$ admits an optimal priority algorithm [12]. This indicates that meaningful characterizations of subproblems admitting optimal priority algorithms may not be possible. In this paper we consider those $\mathbb{S}$ that do admit optimal priority algorithms, and we show that the way in which all such algorithms compute has non-trivial and interesting structural features.