Minimum Activation Cost Edge-Disjoint Paths in Graphs with Bounded Tree-Width

In activation network problems we are given a directed or undirected graph \(G=(V,E)\) with a family \(\{f_{uv} : (u,v)\in E\}\) of monotone non-decreasing activation functions from \(D^2\) to \(\{0,1\}\), where D is a constant-size subset of the non-negative real numbers, and the goal is to find activation values \(x_v \in D\) for all \(v\in V\) of minimum total cost \(\sum _{v \in V}{x_v}\) such that the activated set of edges satisfies some connectivity requirements. We propose an algorithm that optimally solves the minimum activation cost of k edge-disjoint st -paths (st-MAEDP) problem in \(O(|V||D|^{tw+1}tw^3(k+1)^{(tw +3)^{2(tw +3)}}(tw+3)^{2(tw+3)+3})\) time for graphs with treewidth bounded by a constant tw.