Complementing recent progress on classical complexity and polynomial-time approximability of feedback set problems in (bipartite) tournaments, we extend and improve fixed-parameter tractability results for these problems. We show that Feedback Vertex Set in tournaments (FVST) is amenable to the novel iterative compression technique, and we provide a depth-bounded search tree for Feedback Arc Set in bipartite tournaments based on a new forbidden subgraph characterization. Moreover, we apply the iterative compression technique to d-Hitting Set, which generalizes Feedback Vertex Set in tournaments, and obtain improved upper bounds for the time needed to solve 4-Hitting Set and 5-Hitting Set. Using our parameterized algorithm for Feedback Vertex Set in tournaments, we also give an exact (not parameterized) algorithm for it running in O(1.709 n) time, where n is the number of input graph vertices, answering a question of Woeginger [Discrete Appl. Math. 156(3):397–405, 2008].

Abstract. We prove that finding a rooted subtree with at least k 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 that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in 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 D ∈ L of order n with minimum in-degree at least 3 contains a rooted spanning tree, then D contains one with at least (n/2) 1/5 − 1 leaves. 1

Abstract. There are different ways for an external agent to influence the outcome of an election. We concentrate on “control ” by adding or deleting candidates of an election. Our main focus is to investigate the parameterized complexity of various control problems for different voting systems. To this end, we introduce natural digraph problems that may be of independent interest. They help in determining the parameterized complexity of control for different voting systems including Llull, Copeland, and plurality votings. Devising several parameterized reductions, we provide a parameterized complexity overview of the digraph and control problems with respect to natural parameters. 1

Abstract. We give an fpt approximation algorithm for the directed vertex disjoint cycle problem. Given a directed graph G with n vertices and a positive integer k, the algorithm constructs a family of at least k/ρ(k) disjoint cycles of G if the graph G has a family of at least k disjoint cycles (and otherwise may still produce a solution, or just report failure). Here ρ is a computable function such that k/ρ(k) is nondecreasing and unbounded. The running time of our algorithm is polynomial. The directed vertex disjoint cycle problem is hard for the parameterized complexity class W[1], and to the best of our knowledge our algorithm is the first fpt approximation algorithm for a natural W[1]-hard problem. Key words: approximation algorithms, fixed-parameter tractability, parameterized complexity theory. 1

We present the first thorough theoretical analysis of the Transitivity Editing problem on digraphs. Herein, the task is to make a given digraph transitive by a minimum number of arc insertions or deletions. Transitivity Editing has recently been identified as important for the detection of hierarchical structure in molecular characteristics of disease. We present a first proof of NP-hardness, which also extends to the restricted cases where the input digraph is acyclic or has maximum degree three. Moreover, we improve previous fixed-parameter algorithms, now achieving a running time of O(2.57 k + n 3) for an n-vertex digraph if k arc modifications are sufficient to make it transitive. By providing an O(k 2)-vertex problem kernel, we also answer an open question from the literature. In case of digraphs with maximum degree d, an O(k ·d)-vertex problem kernel can be shown. We also demonstrate that if the input digraph does not contain “diamonds”, then there is an optimal solution that performs only arc deletions. Our hardness as well as algorithmic results transfer to Transitivity Deletion, where only arc deletions are allowed.

Abstract. The 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 n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1). 1