An FPT algorithm for node-disjoint subtrees problems parameterized by treewidth
Résumé
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 C1, C2, ..., Cm, D, a set of m integers K1, K2, ..., Km and an integer L, in the search of a forest (T1, T2, ..., TL) containing L node-disjoint trees of G. Each mapping is a function V x N -> Z. For each tree Ti, it associates each node v of V to a score depending only on the degree d_{T_i}(v) of v in Ti (possibly 0 if the node is not in Ti. We then get, for each tree, a set of m + 1 scores, one per mapping, by summing the scores of the nodes. For each tree in the forest and i <= m, the i-th score should be lower than Ki. In addition, the forest should minimize the sum of the scores of all the trees related to the mapping D. We proceed to a parameterized analysis of the \MTDW problem with regard to four parameters that are the number of constraints, the maximum degree after which the constraints and the objective function are constant, the value L, and the treewidth of the input graph G. 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 effet, 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.