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

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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 Ω(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 ε> 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 σ 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 ε, when x, ε, σ and k can be regarded as constants.