Generalized Pseudoforest Deletion: Algorithms and Uniform Kernel Geevarghese Philip1, Ashutosh Rai2, Saket Saurabh2? 1 Max-Planck-Institut f¨urInformatik (MPII), Germany. [email protected] 2 The Institute of Mathematical Sciences, Chennai, India. fashutosh|[email protected] 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 in- teresting 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? 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 com- plexity. We show that l-pseudoforest Deletion admits an algorithm k O(1) 2 with 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 polynomial kernel for l-pseudoforest Deletion. For the special case of l = 1, we design an algorithm with running time 7:5618knO(1). Our al- gorithms and uniform kernels combine iterative compression, 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 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 [10]. These problems are one of the most well studied problems in all those algorithmic paradigms that are meant for ? 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. 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 run- ning 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 in- stance (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 poly- nomial, we say that Q admits a polynomial kernel. The study of kernelization is a major research frontier of parameterized complexity and many important re- cent advances in the area are on kernelization. For more background, the reader is referred to the monographs [3,11]. The Feedback Vertex Set problem has been widely studied in the field of parameterized algorithms. A series of results have improved the running times to O∗(3:619k) in deterministic setting [9] and O∗(3k) in randomized setting [2], where the O∗ notation hides the polynomial factors. 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 made forest by deleting at most l edges. On the other hand the graph class, Fl contains all graphs where each connected component can be made forest by deleting at most l edges. Graphs in Gl are called almost l-forest. 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 [12] 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, Almost Forest Deletion, parameterized by k + l. They obtained a 2O(l+k)nO(1) algorithm 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 algorithms of [12] 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 2 protrusions. Protrusions of a graph are subgraphs which have small boundary and a small treewidth. A protrusion replacer is an algorithm which identifies large protrusions and replaces them with smaller ones. Fomin et al. [6] use protrusion- replacer to arrive at FPT and kernelization results for Planar-F Deletion. We first apply the techniques used in [6] to get an FPT algorithm for l- pseudoforest Deletion. To that end, we have to show that l-pseudoforest Deletion has a protrusion replacer, which we do by showing that the property of being an l-pseudoforest is strongly monotone and minor-closed. We arrive at a ∗ k running time of O (cl ) for l-pseudoforest Deletion where cl is a function of l alone. If we try to apply the machinery of [6] to get a kernelization algorithm for l-pseudoforest Deletion, it only gives a kernel of size kc where the constant c depends on l. We use the similarity of this problem with Feedback Vertex set and apply Gallai's theorem and Expansion Lemma to decrease the maximum degree of the graph. This, when combined with techniques used in [6], gives us a kernel of size ck2, where the constant c depends on l. These kind of kernels are more desired as it gives O(k2) kernel for every fixed l, while the non-uniform kernelization does give a polynomial kernel for every fixed l, but the exponent's dependency on l makes the size of the kernel grow very quickly when compared to uniform-kernelization case. This result is one of the main contributions of the paper and should be viewed as another result similar to the one obtained recently by Giannopoulou et al. [8]. We also looked at a special case for of l-pseudoforest Deletion, namely Pseudoforest Deletion, where we ask whether we can delete at most k vertices to get to a pseudoforest. A pseudoforest is special case of l-pseudoforest for l = 1, i.e. in a pseudoforest, each connected component is just one edge away from being a tree. In other words, it is a class of graphs where every connected component has at most one cycle. We apply the well known technique of iterative compression along with a non-trivial measure and an interesting base case to arrive at an O∗(7:5618k) algorithm for this problem. We also give an explicit kernel with O(k2) vertices for the problem. 2 Preliminaries In this section, we first give the notations and definitions which are used in the paper. Then we state some basic properties about l-pseudoforests and some known results which will be used. Notations and Definitions: For a graph G, we denote the set of vertices of the graph by V (G) and the set of edges of the graph by E(G). We denote jV (G)j and jE(G)j by n and m respectively, where the graph is clear from context. For a set S ⊆ V (G), the subgraph of G induced by S is denoted by G[S] and it is defined as the subgraph of G with vertex set S and edge set f(u; v) 2 E(G): u; v 2 Sg and the subgraph obtained after deleting S is denoted as G − S.