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We study the problem of finding a minimum weight connected subgraph spanning at least $k$ vertices on planar, node-weighted graphs. We give a $(4+eps)$-approximation algorithm for this problem. We achieve this by utilizing the recent LMP primal-dual $3$-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al (SWAT16) 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 $( icefrac{4}{3}cdot r+eps)$-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 (STACS08) 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 emph{non-black-box} fashion could not beat the factor $ icefrac{4}{3}cdot r$ when the tree merging step relies only on the solutions output by the LMP algorithm.
We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a ($2.88
We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph $G=(V,E)$ and integer connectivity requirements $r(uv)$ for each unordered pair of
We study the design of schedules for multi-commodity multicast; we are given an undirected graph $G$ and a collection of source destination pairs, and the goal is to schedule a minimum-length sequence of matchings that connects every source with its
We study the classical Node-Disjoint Paths (NDP) problem: given an $n$-vertex graph $G$ and a collection $M={(s_1,t_1),ldots,(s_k,t_k)}$ of pairs of vertices of $G$ called demand pairs, find a maximum-cardinality set of node-disjoint paths connecting
For any $epsilon>0$, Laue and Matijevi{c} [CCCG07, IPL08] give a PTAS for finding a $(1+epsilon)$-approximate solution to the $k$-hop MST problem in the Euclidean plane that runs in time $(n/epsilon)^{O(k/epsilon)}$. In this paper, we present an algo