We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.

Abstract. It is known to be NP-hard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomial-time algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n k) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices to be deleted is fixed-parameter tractable. This answers an open question of Cai [2]. 1

We examine exact algorithms for the NP-hard Graph Bipartization problem. The task is, given a graph, to find a minimum set of vertices to delete to make it bipartite. Based on the “iterative compression ” method introduced by Reed, Smith, and Vetta in 2004, we present new algorithms and experimental results. The worst-case time complexity is improved. Based on new structural insights, we give a simplified correctness proof. This also allows us to establish a heuristic improvement that in particular speeds up the search on dense graphs. Our best algorithm can solve all instances from a testbed from computational biology within minutes, whereas established methods are only able to solve about half of the instances within reasonable time.

Abstract. We survey the conceptual framework and several applications of the iterative compression technique introduced in 2004 by Reed, Smith, and Vetta. This technique has proven very useful for achieving a number of recent breakthroughs in the development of fixed-parameter algorithms for NP-hard minimization problems. There is a clear potential for further applications as well as a further development of the technique itself. We describe several algorithmic results based on iterative compression and point out some challenges for future research. 1

Abstract. We present a randomized subexponential time, polynomial space parameterized algorithm for the k-Weighted Feedback Arc Set in Tournaments (k-FAST) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kind of universal hash functions, that we coin universal coloring families. For integers m, k and r, a family F of functions from [m] to [r] is called a universal (m, k, r)-coloring family if for any graph G on the set of vertices [m] with at most k edges, there exists an f ∈ F which is a proper vertex coloring of G. Our algorithm is the first non-trivial subexponential time parameterized algorithm outside the framework of bidimensionality. 1

We survey known results on parameterized complexity of the feedback set and induced subdigraph problems for digraphs. We prove new results on some parameterizations of the paired comparison problems on digraphs. One of our theorems implies a new result for a parameterized version of the linear arrangement problem for undirected graphs. We state several open problems. 1

The Feedback Arc Set problem asks whether it is possible to delete at most k arcs to make a directed graph acyclic. We show that Feedback Arc Set is NPcomplete for bipartite tournaments, that is, directed graphs that are orientations of complete bipartite graphs. Key words: combinatorial problems, computational complexity, feedback set problems, bipartite tournaments Given a directed graph G = (V, A) with vertex set V and arc set A, a feedback vertex (or arc) set is a subset of vertices (or arcs) that meets all cycles in G. The Feedback Vertex/Arc Set (FVS/FAS) problems ask to decide, for a given graph G and a nonnegative integer k, whether there is a feedback vertex/arc set of size at most k. Both problems are known to be NP-complete if we put no restriction on the input directed graphs [3]. Motivated by the general hardness results, many subclasses of directed graphs have been considered. It turns out that both FVS and FAS can be solved in polynomial time in reducible flow graphs [10] and in cyclically reducible flow graphs [12]. Due to important applications, e.g. in voting systems [7], feedback set problems restricted to tournaments received considerable attention. A tournament is a directed graph where there is exactly one arc between each

Abstract. To decide if the parameterized feedback vertex set problem in directed graph is fixed-parameter tractable is a long standing open problem. In this paper, we prove that the parameterized feedback vertex set in directed graph is fixed-parameter tractable and give the first FPT algorithm of running time O((1.48k) k n O(1) ) for it. As the feedback arc set problem in directed graph can be transformed to a feedback vertex set problem in directed graph, hence we also show that the parameterized feedback arc set problem can be solved in time of O((1.48k) k n O(1)).

Abstract. We resolve positively a long standing open question regarding the fixed-parameter tractability of the parameterized Directed Feedback Vertex Set problem. In particular, we propose an algorithm which solves this problem in O(8 k k! ∗ poly(n)). Keywords. Directed Feedback Vertex Set, Fixed-Parameter Tractability 1

A tournament T = (V, A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed ɛ> 0, the kernelized instance has at most (2 + ɛ)k vertices. Our result improves the previous known bound of O(k²) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.