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

Abstract

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 ${{{{\varvec{a}}}}}\in \mathbb {Z}_{>0}^{m}$, whose components are given for each edges $e\in E$. 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 $x\in \mathbb {R}_{\ge 0}$, 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 $K_G({{{{\varvec{a}}}}},x)$ 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 $\mathrm{Vol}(K_G({{{{\varvec{a}}}}},x))$ if the pathwidth of G is bounded by a constant p. Our algorithm outputs a value $V'$ satisfying that $1\le V'/\mathrm{Vol}(K_G({{{{\varvec{a}}}}},x)) \le 1+\epsilon$ and finishes in $O(p^{4}2^{1.5p}n(\frac{2mnp}{\epsilon })^{3p}L)$ time, where L is the number of bits in the input. If the pathwidth p is a constant, the running time is $O(n(\frac{mn}{\epsilon })^{3p}L)$.