Minimum activation cost node-disjoint paths in graphs with bounded treewidth

In activation network problems we are given a directed or undirected graph G = (V,E) with a family {f uv : (u,v) ∈ E} of monotone non-decreasing activation functions from D2 to {0,1}, where D is a constant-size subset of the non-negative real numbers, and the goal is to find activation values xv for all v ∈ V of minimum total cost ∑  v ∈ V x v such that the activated set of edges satisfies some connectivity requirements. We propose algorithms that optimally solve the minimum activation cost of k node-disjoint st-paths (st-MANDP) problem in O(tw ((5 + tw)|D|)2tw + 2|V|3) time and the minimum activation cost of node-disjoint paths (MANDP) problem for k disjoint terminal pairs (s 1,t 1),…,(s k ,t k ) in O(tw ((4 + 3tw)|D|)2tw + 2|V|) time for graphs with treewidth bounded by tw.