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

TL;DR

This paper develops a polynomial-time approximation scheme for the distribution of the longest path length in DAGs with bounded treewidth and random edge lengths, extending computational feasibility beyond known hard cases.

Contribution

It introduces a fully polynomial time approximation scheme for the longest path distribution in bounded treewidth DAGs, and provides exact algorithms for exponential edge lengths.

Findings

FPTAS achieves (1+ε) approximation ratio with polynomial time for fixed treewidth.

Exact algorithm for exponential distributions runs in time polynomial in graph size and treewidth.

Computational complexity is manageable for graphs with small treewidth and certain distribution conditions.

Abstract

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.