Approximating Node-Weighted k-MST on Planar Graphs

We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4 + e)-approximation algorithm for this problem. We achieve this by utilizing the recent Lagrangian-multiplier preserving (LMP) primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al. (SWAT’16) and adopting an approach by Chudak et al. (Math. Prog. ’04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math. Prog. ’06). More generally, our approach readily gives a (4/3 ⋅ r + e)-approximation on any graph class where the algorithm of Byrka et al. for the prize-collecting version gives an r-approximation. We argue that this can be interpreted as a generalization of an analogous result by Konemann et al. (Algorithmica ’11) for partial cover problems. Together with a lower bound construction by Mestre (STACS’08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al. in a non-black-box fashion could not beat the factor 4/3 ⋅ r when the tree merging step relies only on the solutions output by the LMP algorithm.