An FPTAS for Computing the Distribution Function of the Longest Path Length in DAGs with Uniformly Distributed Edge Lengths

Given a directed acyclic graph ( DAG) G = ( V, E) with n vertices and m edges, we consider random edge lengths. That is, as the input, we have a epsilon Z(>)(m)0, whose components are given for each edges e epsilon E. Then, the random length Y-e of edge e is a mutually independent random variable that obeys a uniform distribution on [ 0, a(e)]. In this paper, we consider the probability that the longest path length is at most a certain value x epsilon R >= 0, which is equal to the probability that all paths in G have length at most x. The problem can be considered as the computation of an m-dimensional polytope K-G (a, x) that is a hypercube truncated by exponentially many hyperplanes that are as many as the number of paths in G. This problem is #P-hard even if G is a directed path. In this paper, motivated by the recent technique of deterministic approximation of #P-hard problems, we show that there is a deterministic FPTAS for the problem of computing Vol(K-G( a, x)) if the pathwidth of G is bounded by a constant p. Our algorithm outputs a value V ' satisfying that 1 <= V '/Vol(K-G(a, x)) <= 1 + epsilon and finishes in O( p(4)2(1.5p)n(2mnp/epsilon)(3p) L) time, where L is the number of bits in the input. If the pathwidth p is a constant, the running time is O(n(mn/epsilon)L-3p).