Two-Processor Scheduling with Start-Times and Deadlines

Abstract

Given a set $\mathcal{T} = \{ T_1 ,T_2 , \cdots ,T_n \} $ of tasks, each $T_i$ having execution time 1, an integer start-time $s_i \geqq 0$ and a deadline $d_i > 0$, along with precedence constraints among the tasks, we examine the problem of determining whether there exists a schedule on two identical processors that executes each task in the time interval between its start-time and deadline. We present an $O(n^3)$ algorithm that constructs such a schedule whenever one exists. The algorithm may also be used in a binary search mode to find the shortest such schedule or to find a schedule that minimizes maximum “tardiness”. A number of natural extensions of this problem are seen to be $NP$ complete and hence probably intractable.