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A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T - S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w: V (T) → N, and the task is to find a feedback vertex set S in T minimizing w(S) = Σ ν ∈S w(ν). Rounding optimal solutions to the natural LPrelaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time. [ABSTRACT FROM AUTHOR]

FEEDBACK VERTEX SET (FVS) is one of the most well-studied problems in the realm of parameterized complexity. In this problem we are given a graph G and a positive integer k and the objective is to test whether there exists S ⊆ V (G) of size at most k such that G - S is a forest. Thus, FVS is about deleting as few vertices as possible to get a forest. The main goal of this paper is to study the following interesting problem: How can we generalize the family of forests such that the nice structural properties of forests and the interesting algorithmic properties of FVS can be extended to problems on this class? Toward this we define a graph class, F1, that contains all graphs where each connected component can be transformed into a forest by deleting at most l edges. A graph in the class F1 is known as pseudoforest in the literature and we call a graph in Fl an l-pseudoforest. We study the problem of deleting k vertices to get into Fl, l-PSEUDOFOREST DELETION, in the realm of parameterized complexity. We show that l-PSEUDOFOREST DELETION admits an algorithm with running time clknO(1) and admits a kernel of size f(l)k2. Thus, for every fixed l we have a kernel of size O(k2). That is, we get a uniform polynomial kernel for l-pseudoforest Deletion. Our algorithms and uniform kernels involve the use of the expansion lemma and protrusion machinery. [ABSTRACT FROM AUTHOR]

The input to the NP-hard POINT LINE COVER problem (PLC) consists of a set P of n points on the plane and a positive integer k; the question is whether there exists a set of at most k lines that pass through all points in P. By straightforward reduction rules, one can efficiently reduce any input to one with at most k2 points. We show that this easy reduction is already essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, for any ε > 0, there is no polynomial-time algorithm that reduces every instance (P, k) of PLC to an equivalent instance with O(k2-ε) points. This answers, in the negative, an open problem posed by Lokshtanov [2009]. Our proof uses the notion of a kernel from parameterized complexity, and the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek [2010, 2014]. It has two main ingredients: We first show, by reduction from VERTEX COVER, that--unless the polynomial hierarchy collapses--PLC has no kernel of total size O(k2-ε) bits. This does not directly imply the claimed lower bound on the number of points, since the best-known polynomialtime encoding of a PLC instance with n points requires ω(n²) bits. To get around this hurdle, we build on work of Alon [1986] and devise an oracle communication protocol of cost O(nlog n) for PLC. This protocol, together with the lower bound on the total size (which also holds for such protocols), yields the stated lower bound on the number of points. While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is--to the best of our knowledge--the first to show a nontrivial lower bound for structural/secondary parameters. It is also the first example of a lower bound for kernelization that makes use of the full power of the oracle communication protocol lower bounds that can be obtained from the work of Dell and van Melkebeek. We combine the main abstract ideas of our proof to derive a general recipe that could be used to obtain such lower bounds for other problems with unknown or insufficiently strong encodings. [ABSTRACT FROM AUTHOR]

We study a general class of problems called F-DELETION problems. In an F-DELETION problem, we are asked whether a subset of at most k vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We study the problem parameterized by k, using p-F-DELETION to refer to the parameterized version of the problem. We obtain a number of algorithmic results on the p-F-DELETION problem when F contains a planar graph. We give a linear vertex kernel on graphs excluding t-claw K1,t, the star with t leaves, as an induced subgraph, where t is a fixed integer and an approximation algorithm achieving an approximation ratio of O(log3/2 OPT), where OPT is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F only contains graph θc as a minor for a fixed integer c. The graph θc consists of two vertices connected by c parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as VERTEX COVER, FEEDBACK VERTEX SET, and DIAMOND HITTING SET. The generic kernelization algorithm is based on a nontrivial application of protrusion techniques, previously used only for problems on topological graph classes. [ABSTRACT FROM AUTHOR]

According to the classical Erdős-Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most [ABSTRACT FROM AUTHOR]

We show that for every fixed j ≥ i ≥ 1, the k-DOMINATING SET problem restricted to graphs that do not have Ki,j (the complete bipartite graph on (i + j) vertices, where the two parts have i and j vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a Ki,j-free graph G and a nonnegative integer k, constructs a graph H (the "kernel") and an integer k' such that (1) G has a dominating set of size at most k if and only if H has a dominating set of size at most k', (2) H has O((j+1)i+1ki2) vertices, and (3) k' = O((j+1)i+1ki2). Since d-degenerate graphs do not have Kd+1,d+1 as a subgraph, this immediately yields a polynomial kernel on O((d+2)d+2k(d+1)2) vertices for the k-DOMINATING SET problem on d-degenerate graphs, solving an open problem posed by Alon and Gutner [Alon and Gutner 2008; Gutner 2009]. The most general class of graphs for which a polynomial kernel was previously known for κ-DOMINATING SET is the class of Kh-topological-minor-free graphs [Gutner 2009]. Graphs of bounded degeneracy are the most general class of graphs for which an FPT algorithm was previously known for this problem. Kh-topologicalminor- free graphs are Ki,j-free for suitable values of i, j (but not vice-versa), and so our results show that κ-DOMINATING SET has both FPT algorithms and polynomial kernels in strictly more general classes of graphs. Using the same techniques, we also obtain an O(jki) vertex-kernel for the k-INDEPENDENT DOMINATING SET problem on Ki,j-free graphs. [ABSTRACT FROM AUTHOR]

In this work, we study the d-Hitting Set and Feedback Vertex Set problems through the paradigm of finding diverse collections of r solutions of size at most k each, which has recently been introduced to the field of parameterized complexity. This paradigm is aimed at addressing the loss of important side information which typically occurs during the abstraction process that models real-world problems as computational problems. We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the minimum pairwise Hamming distance. We show that both problems are fixed-parameter tractable in k + r for both diversity measures. A key ingredient in our algorithms is a (problem independent) network flow formulation that, given a set of 'base' solutions, computes a maximally diverse collection of solutions. We believe that this could be of independent interest. [ABSTRACT FROM AUTHOR]