Probabilistic Bounds on the Performance of List Scheduling

Abstract

The problem of scheduling tasks on m processors to minimize the schedule length (makespan) is NP-complete. Here we study the behavior of list schedules under the assumptions that there are no task precedence constraints and that task times are chosen from a uniform distribution. We show that, given a desired degree of confidence $1 - \varepsilon $, we can find a minimum sample size N such that if $n \geqq N$ and the n task times $\bar X = (X_1 , \cdots ,X_n )$ are chosen from any uniform distribution, then \[ {\bf P}\left[ {\frac{{L(\bar X)}}{{{\operatorname{OPT}}(\bar X)}} < 1 + \frac{{4(m - 1)}}{n}} \right] > 1 - \varepsilon \] where $L(\bar X)$ is the length of any list schedule and ${\operatorname{OPT}}(\bar X)$ is the length of the optimal schedule. Thus for n sufficiently large, the performance of any list schedule can be made arbitrarily close to that of the optimal policy with any desired degree of confidence. For example, for $m = 2$ and $\varepsilon = 0.01$, the ratio is bounded by 1.11 when $n = 36$ and bounded by 1.03 when $n = 100$.