I would like to cite a paper/report/etc that solves the following problem polynomially in $n$:Given a weighted directed graph $G=(V,E)$, $|V|=n$, of bounded treewidth $k \in \mathbb{N}$ and a source-destination pair $s,t\in V$, find a longest path (not walk) from $s$ to $t$.The corresponding Wikipedia article (https://en.wikipedia.org/wiki/Longest_path_problem#Parameterized_complexity)  implies that it is possible: the longest path problem is [...] fixed-parameter tractable when parameterized by the treewidth of the graphSadly, my knowledge of treewidth-techniques is rather small.I found the following paper, but it is on undirected (?unweighted?) graphs: Confronting Hardness Using a Hybrid Approach: Cites a paper of Bodlaender (in Section 2.2.1), where Theorem 2.2 states that the longest path problem is solvable in $O(2^k k! n)$ (assuming a treewidth decomposition is given)Is it easy to see that this technique also extends to directed weighted graphs? What happens when the weights are encoded in binary, i.e., they can be exponential in $n$?There is also a nice discussion here, but it also seems to rely on undirected (?unweighted?) graphs.Thank you!
Paper regarding the complexity of the longest path problem on weighted directed graphs of bounded treewidth
cc.complexity theory;graph algorithms;treewidth;fixed parameter tractable;dynamic programming