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 \(O(n(\frac{mn}{\epsilon })^{3p}L)\).