Computing the Exact Distribution Function of the Stochastic Longest Path Length in a DAG

Abstract

Consider the longest path problem for directed acyclic graphs (DAGs), where a mutually independent random variable is associated with each of the edges as its edge length. Given a DAG G and any distributions that the random variables obey, let F MAX (x ) be the distribution function of the longest path length. We first represent F MAX (x ) by a repeated integral that involves n *** 1 integrals, where n is the order of G . We next present an algorithm to symbolically execute the repeated integral, provided that the random variables obey the standard exponential distribution. Although there can be *** (2 n ) paths in G , its running time is bounded by a polynomial in n , provided that k , the cardinality of the maximum anti-chain of the incidence graph of G , is bounded by a constant. We finally propose an algorithm that takes x and *** > 0 as inputs and approximates the value of repeated integral of x , assuming that the edge length distributions satisfy the following three natural conditions: (1) The length of each edge (v i ,v j ) *** E is non-negative, (2) the Taylor series of its distribution function F ij (x ) converges to F ij (x ), and (3) there is a constant *** that satisfies $\sigma^p \le \left|\left(\frac{d}{dx}\right)^p F_{ij}(x)\right|$ for any non-negative integer p . It runs in polynomial time in n , and its error is bounded by *** , when x , *** , *** and k can be regarded as constants.