A Generic Algorithm for Approximately Solving Stochastic Graph Optimization Problems

Abstract

Given a (directed or undirected) graph G = (V,E), a mutually independent random variable X _e obeying a normal distribution for each edge e ∈ E that represents its edge weight, and a property $\cal P$ on graph, a stochastic graph maximization problem asks the distribution function F _MAX(x) of random variable $X_{\rm MAX} = \max_{P \in {\cal P}} \sum_{e \in A} X_e$, where property ${\cal P}$ is identified with the set of subgraphs P = (U,A) of G having $\cal P$. This paper proposes a generic algorithm for computing an elementary function $\tilde F(x)$ that approximates F _MAX(x). It is applicable to any $\cal P$ and runs in time $O(T_{A_{\rm MAX}} ({\cal P})+ T_{A_{\rm CNT}} ({\cal P}))$, provided the existence of an algorithm A _MAX that solves the (deterministic) graph maximization problem for $\cal P$ in time $T_{A_{\rm MAX}} ({\cal P})$ and an algorithm A _CNT that outputs an upper bound on $|{\cal P}|$ in time $T_{A_{\rm CNT}} ({\cal P})$. We analyze the approximation ratio and apply it to three graph maximization problems. In case no efficient algorithms are known for solving the graph maximization problem for $\cal P$, an approximation algorithm A _APR can be used instead of A _MAX to reduce the time complexity, at the expense of increase of approximation ratio. Our algorithm can be modified to handle minimization problems.