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

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 $O(k^2 n(\frac{(4k+2)mn}{\epsilon})^{4k^2+5k+1})$ 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 $((4k+2)mn)^{O(k)}$ 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.