17,412 research outputs found

    A 2k2k-Vertex Kernel for Maximum Internal Spanning Tree

    Full text link
    We consider the parameterized version of the maximum internal spanning tree problem, which, given an nn-vertex graph and a parameter kk, asks for a spanning tree with at least kk internal vertices. Fomin et al. [J. Comput. System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a 3k3k-vertex kernel. Here we propose a novel way to use the same reduction rule, resulting in an improved 2k2k-vertex kernel. Our algorithm applies first a greedy procedure consisting of a sequence of local exchange operations, which ends with a local-optimal spanning tree, and then uses this special tree to find a reducible structure. As a corollary of our kernel, we obtain a deterministic algorithm for the problem running in time 4knO(1)4^k \cdot n^{O(1)}

    Max-Leaves Spanning Tree is APX-hard for Cubic Graphs

    Full text link
    We consider the problem of finding a spanning tree with maximum number of leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a 3/2-approximation algorithm when restricted to graphs where every vertex has degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic graphs. The APX-hardness of the related problem Minimum Connected Dominating Set for cubic graphs follows

    Distributed Connectivity Decomposition

    Full text link
    We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity kk into (fractionally) vertex-disjoint weighted dominating trees with total weight Ω(klogn)\Omega(\frac{k}{\log n}), in O~(D+n)\widetilde{O}(D+\sqrt{n}) rounds. (II) A decomposition of each undirected graph with edge-connectivity λ\lambda into (fractionally) edge-disjoint weighted spanning trees with total weight λ12(1ε)\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon), in O~(D+nλ)\widetilde{O}(D+\sqrt{n\lambda}) rounds. We also show round complexity lower bounds of Ω~(D+nk)\tilde{\Omega}(D+\sqrt{\frac{n}{k}}) and Ω~(D+nλ)\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}}) for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from O(n3)O(n^3) to near-optimal O~(m)\tilde{O}(m). As corollaries, we also get distributed oblivious routing broadcast with O(1)O(1)-competitive edge-congestion and O(logn)O(\log n)-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal O(logn)O(\log n)-approximation of vertex connectivity: centralized O~(m)\widetilde{O}(m) and distributed O~(D+n)\tilde{O}(D+\sqrt{n}). The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an O(m)O(m) centralized exact algorithm while the latter is the first distributed vertex connectivity approximation

    Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition

    Full text link
    We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now Lemma 5.2, as the previous proof was erroneou
    corecore