Generalized Pseudoforest Deletion: Algorithms and Uniform Kernel? Geevarghese Philip1, Ashutosh Rai2, Saket Saurabh2;3?? 1 Chennai Mathematical Institute, Chennai, India. [email protected] 2 The Institute of Mathematical Sciences, HBNI, Chennai, India. fashutosh|[email protected] 3 University of Bergen, Bergen, Norway. Abstract. Feedback Vertex Set (FVS) is one of the most well stud- ied 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 proper- ties of FVS can be extended to problems on this class? Towards this we define a graph class, Fl, that contains all graphs where each connected component can transformed into forest by deleting at most l edges. The class F1 is known as pseudoforest in the literature and we call Fl as l-pseudoforest. We study the problem of deleting k vertices to get into Fl, l-pseudoforest Deletion, in the realm of parameterized complex- ity. We show that l-pseudoforest Deletion admits an algorithm with k O(1) 2 running time cl n and admits a kernel of size f(l)k . Thus, for every fixed l we have a kernel of size O(k2). That is, we get a uniform polyno- mial kernel for l-pseudoforest Deletion. Our algorithms and uniform kernels involve the use of expansion lemma and protrusion machinery. 1 Introduction In the field of graph algorithms, vertex deletion problems constitute a consid- erable fraction. In these problems we need to delete a small number of vertices ? A preliminary version of this paper appeared at 40th International Symposium on Mathematical Foundations of Computer Science (MFCS) held at Milano, Italy on August 24-28, 2015. The current version contains all the proofs which were omitted from the conference version. Conference version of this paper contained an algorithm for Pseudoforest Deletion based on iterative compression technique which runs in time O∗(7:5618k). This was later improved to O(3knkO(1)) by Bodlaender et al. [2]. Hence we have omitted describing the O∗(7:5618k) algorithm in the current version. ?? The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 306992. such that the resulting graph satisfies certain properties. Many well known prob- lems like Vertex Cover and Feedback Vertex Set (given a graph G and a positive integer k, does there exists S ⊆ V (G) of size at most k such that G − S is a forest) fall under this category. Most of these problems are NP-complete due to a classic result by Lewis and Yannakakis [15]. These problems are one of the most well studied problems in all those algorithmic paradigms that are meant for coping with NP-hardness, such as approximation algorithms and parameterized complexity. The topic of this paper is a generalization of Feedback Vertex Set in the realm of parameterized complexity. The field of parameterized complexity tries to provide efficient algorithms for NP-complete problems by going from the classical view of single-variate measure of the running time to a multi-variate one. It aims at getting algorithms of running time f(k)nO(1), where k is an integer measuring some aspect of the problem. These algorithms are called fixed parameter tractable (FPT) algorithms and the integer k is called the parameter. In most of the cases, the solution size is taken to be the parameter, which means that this approach gives faster algorithms when the solution is of small size. It is known that a decidable problem is FPT if and only if it is kernelizable: a kernelization algorithm for a problem Q takes an instance (x; k) and in time polynomial in jxj+k produces an equivalent instance (x0; k0) (i.e., (x; k) 2 Q iff (x0; k0) 2 Q) such that jx0j + k0 ≤ g(k) for some computable function g. The function g is the size of the kernel, and if it is polynomial, we say that Q admits a polynomial kernel. The study of kernelization is a major research frontier of parameterized complexity and many important recent advances in the area are on kernelization. For more background, the reader is referred to the monographs [7,16]. The Feedback Vertex Set problem has been widely studied in the field of parameterized algorithms. The earlier algorithms used the branch and bound approach, and the first O∗(ck) algorithm (here the O∗ notation hides the poly- nomial factors) was obtained by Guo et al. [13] using the iterative compression technique, that was introduced by Reed et al. [19] to solve the Odd Cycle Transversal problem. After this, a series of results have improved the run- ning times to O∗(3:619k) in deterministic setting [14] and O∗(3k) in randomized setting [5]. The kernelization question for Feedback Vertex Set was open for a long time, and was resolved by Burrage et al. [4] to give a kernel with O(k11) vertices. It was subsequently improved by Bodlaender et al. [3] to a kernel with O(k3) vertices and then by Thomass´e[21] to a kernel with O(k2) vertices. Thomass´e'stechniques included a very clever use of an old result of Gallai [11], which helped in identifying a small set of vertices which hit all the cycles going through neighbourhood of a high degree vertex. After identifying these vertices, edges are put between the high degree vertex v and these impor- tant vertices, and some edges are removed between v and its neighbourhood. This helps in bounding the degree of the graph, which then yields the quadratic ker- nel via a simple counting. It is known that under standard complexity-theoretic assumptions, Feedback Vertex Set has no kernel with O(k(2−")) edges, for 2 any " > 0 [6]. However, it is open whether Feedback Vertex Set has a kernel with O(k(2−")) vertices. 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? There are two ways of quantitatively generalizing forests: given a positive integer l we define graph classes Gl and Fl. The graph class Gl is defined as those graphs that can be transformed into a forest by deleting at most l edges. On the other hand the graph class, Fl contains all graphs where each connected component can be transformed into a forest by deleting at most l edges. Graphs in Gl are called almost l-forest [18]. The class F1 is known as pseudoforest in the literature and we call Fl as l-pseudoforest. In this paper we study the problem of deleting k-vertices to get into Fl, l-pseudoforest Deletion, in the realm of parameterized complexity. Recently, a subset of authors of this work looked at a generalization of the Feedback Vertex Set problem in terms of Gl. In particular they studied the problem of deleting k-vertices to get into Gl, which they called Almost For- est Deletion, parameterized by k + l [18]. They obtained an algorithm with with running time 2O(l+k)nO(1) and a kernel of size O(kl(k + l)). One property of almost-l-forests which is crucial in the design of FPT and kernelization algo- rithms of [18] is that any almost-l-forests on n vertices can have at most n+l −1 edges. The same can not be said about l-pseudoforests and they can turn out to be significantly more dense. So while the techniques used for arriving at FPT and kernelization results for Feedback Vertex Set give similar results for Almost Forest Deletion, they break down when applied directly to l-pseudoforest Deletion. So we had to get into the theory of protrusions. Protrusions of a graph are subgraphs which have small boundary and a small treewidth. A pro- trusion replacer is an algorithm which identifies large protrusions and replaces them with smaller ones. Fomin et al. [10] use protrusion-replacer to arrive at FPT and kernelization results for Planar-F Deletion. For getting an FPT algorithm using the techniques of [10], we first need to show that l-pseudoforest Deletion has a protrusion replacer. We observe that the property of a graph being an l-pseudoforest is minor-closed and hence is CMSO-expressible. This, along with the property of strong monotonicity implies that the problem has a protrusion replacer due to the results of Bodlaneder et al. [1]. Now, along the lines of [10], using the protrusion replacer, we reduce the graph and give a covering guarantee on any solution of the resulting graph. That is, we show that any solution to the resulting graph has to have some fraction of the total number of edges incident on them, where the exact fraction depends only on l and is independent of the solution size. This result is then combined with the bucketing technique of [10] to get an algorithm with running ∗ k time O (cl ) for l-pseudoforest Deletion where cl is a function of l alone.