245 research outputs found

    Parameterized Algorithms for Directed Maximum Leaf Problems

    Full text link
    We prove that finding a rooted subtree with at least kk leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family L\cal L that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in L\cal L. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph DLD\in \cal L of order nn with minimum in-degree at least 3 contains a rooted spanning tree, then DD contains one with at least (n/2)1/51(n/2)^{1/5}-1 leaves

    Spanning directed trees with many leaves

    Get PDF
    The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that - every strongly connected nn-vertex digraph DD with minimum in-degree at least 3 has an out-branching with at least (n/4)1/31(n/4)^{1/3}-1 leaves; - if a strongly connected digraph DD does not contain an out-branching with kk leaves, then the pathwidth of its underlying graph UG(DD) is O(klogk)O(k\log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k4k. The last result implies that it can be decided in time 2O(klog2k)nO(1)2^{O(k\log^2 k)}\cdot n^{O(1)} whether a strongly connected digraph on nn vertices has an out-branching with at least kk leaves. On acyclic digraphs the running time of our algorithm is 2O(klogk)nO(1)2^{O(k\log k)}\cdot n^{O(1)}

    An FPT Algorithm for Directed Spanning k-Leaf

    Get PDF
    An out-branching of a directed graph is a rooted spanning tree with all arcs directed outwards from the root. We consider the problem of deciding whether a given directed graph D has an out-branching with at least k leaves (Directed Spanning k-Leaf). We prove that this problem is fixed parameter tractable, when k is chosen as the parameter. Previously this was only known for restricted classes of directed graphs. The main new ingredient in our approach is a lemma that shows that given a locally optimal out-branching of a directed graph in which every arc is part of at least one out-branching, either an out-branching with at least k leaves exists, or a path decomposition with width O(k^3) can be found. This enables a dynamic programming based algorithm of running time 2^{O(k^3 \log k)} n^{O(1)}, where n=|V(D)|.Comment: 17 pages, 8 figure

    Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

    Get PDF
    We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc kk-Leaf Out-Branching}, which is to find an oriented spanning tree with at least kk leaves, we obtain an algorithm solving the problem in time 2O(klogk)n+nO(1)2^{O(\sqrt{k} \log k)} n+ n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed graph HH as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O(k)n+nO(1)2^{O(\sqrt{k})}n + n^{O(1)}. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc kk-Internal Out-Branching}, which is to find an oriented spanning tree with at least kk internal vertices. We obtain an algorithm solving the problem in time 2O(klogk)+nO(1)2^{O(\sqrt{k} \log k)} + n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed apex graph HH as a minor. Finally, we observe that for any ϵ>0\epsilon>0, the {\sc kk-Directed Path} problem is solvable in time O((1+ϵ)knf(ϵ))O((1+\epsilon)^k n^{f(\epsilon)}), where ff is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

    How to Secure Matchings Against Edge Failures

    Get PDF
    Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We show that this problem is equivalent to covering a digraph with non-trivial strongly connected components at minimal cost. We provide efficient exact and approximation algorithms for this task. In particular, for the unit-cost problem, we give a log_2 n-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we give tight upper and lower approximation bounds relative to the Directed Steiner Forest problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. To obtain our results, we exploit a close relation to the classical Strong Connectivity Augmentation problem as well as directed Steiner problems

    07281 Abstracts Collection -- Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs

    Get PDF
    From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
    corecore