The Distribution Function of the Longest Path Length in Constant Treewidth DAGs with Random Edge Length

• Published in: arXiv.org
• Main Author: Ando, Ei
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 27.03.2022
• Subjects: Algorithms; Approximation; Distribution functions; Hypercubes; Mathematical analysis; Parameters; Polynomials; Probability distribution functions; Run time (computers)
• ISSN: 2331-8422

This paper is about the length \(X_{\rm MAX}\) of the longest path in directed acyclic graph (DAG) \(G=(V,E)\) with random edge lengths, where \(|V|=n\) and \(|E|=m\). When the edge lengths are mutually independent and uniformly distributed, the problem of computing the distribution function \(\Pr[X_{\rm MAX}\le x]\) is known to be \(\#\)P-hard even in case \(G\) is a directed path. In this case, \(\Pr[X_{\rm MAX}\le x]\) is equal to the volume of the knapsack polytope, an \(m\)-dimensional unit hypercube truncated by a halfspace. In this paper, we show that there is a deterministic fully polynomial time approximation scheme (FPTAS) for computing \(\Pr[X_{\rm MAX}\le x]\) in case the treewidth of \(G\) is at most a constant \(k\). The running time of our algorithm is \(O(k^2 n(\frac{16(k+1)mn^2}{\epsilon})^{4k^2+6k+2})\) to achieve a multiplicative approximation ratio \(1+\epsilon\). Before our FPTAS, we present a fundamental formula that represents \(\Pr[X_{\rm MAX}\le x]\) by at most \(n-1\) repetitions of definite integrals. Moreover, in case the edge lengths follow the mutually independent standard exponential distribution, we show a \(((4k+2)mn)^{O(k)}\) time exact algorithm. For random edge lengths satisfying certain conditions, we also show that computing \(\Pr[X_{\rm MAX}\le x]\) is fixed parameter tractable if we choose treewidth \(k\), the additive error \(\epsilon'\), and \(x\) as the parameters.