Estimating the Probability of Meeting a Deadline in Hierarchical Plans

TL;DR

This paper introduces a polynomial-time deterministic algorithm to estimate the probability of meeting a deadline in hierarchical plans with uncertain task durations, addressing an NP-hard problem with practical approximation bounds.

Contribution

It presents the first polynomial-time approximation algorithm with formal bounds for deadline probability estimation in hierarchical plans, along with new complexity results.

Findings

The algorithm outperforms sampling and exact methods in accuracy and speed.

Empirical results show the bounds are often tighter than theoretical guarantees.

The method is scalable with plan size and task duration support.

Abstract

Given a hierarchical plan (or schedule) with uncertain task times, we propose a deterministic polynomial (time and memory) algorithm for estimating the probability that its meets a deadline, or, alternately, that its {\em makespan} is less than a given duration. Approximation is needed as it is known that this problem is NP-hard even for sequential plans (just, a sum of random variables). In addition, we show two new complexity results: (1) Counting the number of events that do not cross deadline is \#P-hard; (2)~Computing the expected makespan of a hierarchical plan is NP-hard. For the proposed approximation algorithm, we establish formal approximation bounds and show that the time and memory complexities grow polynomially with the required accuracy, the number of nodes in the plan, and with the size of the support of the random variables that represent the durations of the primitive tasks. We examine these approximation bounds empirically and demonstrate, using task networks taken from the literature, how our scheme outperforms sampling techniques and exact computation in terms of accuracy and run-time. As the empirical data shows much better error bounds than guaranteed, we also suggest a method for tightening the bounds in some cases.