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) $G=(V,E)$ 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.