This paper is about the length $X_{\rm MAX}$ of the longest path in directed acyclic graph (DAG) $G=(V,E)$ that have random edge lengths, where $|V|=n$ and $|E|=m$. Especially, 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. This is because $\Pr[X_{\rm MAX}\le x]$ 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 {\em deterministic} fully polynomial time approximation scheme (FPTAS) for computing $\Pr[X_{\rm MAX}\le x]$ in case the treewidth of $G$ is bounded by a constant $k$, where there may be exponentially many $s-t$ paths in $G$. The running time of our algorithm is $O((3k+2)^2 n(\frac{(6k+6)mn}{\epsilon})^{9k^2+15k+6})$ to achieve a multiplicative approximation ratio $1+\epsilon$. On the way to show our FPTAS, we show a fundamental formula that represents $\Pr[X_{\rm MAX}\le x]$ by at most $n-1$ repetition of definite integrals. This also leads us to more results. In case the edge lengths obey the mutually independent standard exponential distribution, we show that there exists a $((6k+4)mn)^{O(k)}$ time exact algorithm. We also show, for random edge lengths satisfying certain conditions, 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.
翻译:这张纸是关于在定向环球图(DAG) $G=(V,E) 中程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程程距程程程程程程程程程程程程程程程程程(FTTTAS程程程程程程程程程程程程程程程程程程,且马路路路程路程路程路程路程可显示,即直路路路路路路路路路路程距距距距距距距距距维程距距维程距距距次程距距次程程程程距次程距距距距距距距距程距程程程程程程程程程程程程程程程程程程程程程程程程程。