Convex Stochastic Bounds and Stochastic Optimisation on Graphs

This paper presents an approach to provide stochastic bounds for a large class of optimisation problems on graphs when the parameters (i.e. costs, weights or delays) for links are random variables. We consider the class of problems which are based on convex operators and whose complexity is polynomial, when the parameters are deterministic. Here, the parameters (for instance the delay of a link) are discrete random variables. Such an assumption drastically changes the complexity of the problem (typically, the problems turn out unfortunately to be NP-complete). We propose to give stochastic bounds (both upper and lower bounds) based on convex order. First, we prove how we can simplify a discrete distribution to obtain bounding distributions which are easier to deal with, leading to a tradeoff between the computation complexity and the accuracy of the bounds. Second, we design a polynomial time algorithm to compute an upper bound. The approach is illustrated by the computation of the execution time of a task graph.