An FPT algorithm for node-disjoint subtrees problems parameterized by treewidth

In this paper, we introduce a problem called MINIMUM SUBTREE PROBLEM WITH DEGREE WEIGHTS, or MTDW. This problem generalized covering tree problems like SPANNING TREE, STEINER TREE, MINIMUM BRANCH VERTICES, MINIMUM LEAF SPANNING TREE, or PRIZE COLLECTING STEINER TREE. It consists, given an undirected graph G = (V, E), a set of m + 1 mappings C-1, C-2,...,C-m, D : V x N -> Z, a set of m integers K-1, K-2,...,K-m is an element of Z and a positive integer l, in the search of a forest (T-1,T-2,...,T-l) containing l node-disjoint trees of G. Along with K-l, the mapping C-l defines a constraint that should be satisfied by the trees of the forest. For each tree T-l, it associates each node n of V to the score C-l (upsilon,dT(l)(upsilon)) where d(Tl) (upsilon) is the degree of upsilon in T-l (possibly 0 if the node is not in T-l). The sum Sigma(upsilon is an element of V) C-j(upsilon, d(Tt)(upsilon)) should not exceed K-j. In addition, the forest should minimize Sigma(l)(t=1) Sigma(upsilon is an element of V) D(upsilon,dT(l) (upsilon)). We proceed to a parameterized analysis of the MTDW problem with regard to four parameters that are the number of constraints m, the value l, the treewidth of the input graph G and Delta, the minimum degree above which all the constraints and D are constant (for every j is an element of[1,m], upsilon is an element of V and d >= Delta, C-j(upsilon,d) = C-j(upsilon,Delta), and D(upsilon,d) = D(upsilon,Delta)). For this problem, we provide a first dichotomy P versus NP-hard depending whether the previous parameters are fixed to be constant or not and a second dichotomy FPT versus W[1]-hard depending whether each of these parameters is constant, considered as a parameter, or disregard. As a side effect, we obtained parameterized algorithms, previously undescribed, for problems such that BUDGET STEINER TREE PROBLEM WITH PROFITS, MINIMUM BRANCH VERTICES, GENERALIZED BRANCH VERTICES, or k-BOTTLENECK STEINER TREE.