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Strong Parameterized Deletion: Bipartite Graphs

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Strong Parameterized Deletion: Bipartite Graphs Ashutosh Rai1, M. S. Ramanujan2, and Saket Saurabh∗1,3 1 Institute of Mathematical Sciences, India. {ashutosh,saket}@imsc.res.in 2 Vienna Institute of Technology, Austria. [email protected] 3 University of Bergen, Bergen, Norway. Abstract The purpose of this article is two fold: (a) to formally introduce a stronger version of graph deletion problems; and (b) to study this version in the context of bipartite graphs. Given a family of graphs F, a typical instance of parameterized graph deletion problem consists of an undirected graph G and a positive integer k and the objective is to check whether we can delete at most k vertices (or k edges) such that the resulting graph belongs to F. Another version that has been recently studied is the one where the input contains two integers k1 and k2 and the objective is to check whether we can delete at most k1 vertices and k2 edges such that the resulting graph belongs to F. In this paper, we propose and initiate the study of a more general version which we call strong deletion. In this problem, given an undirected graph G and positive integers k1 and k2, the objective is to check whether there exists a vertex subset S of size at most k1 such that each connected component of G−S can be transformed into a graph in F by deleting at most k2 edges. In this paper we study this stronger version of deletion problems for the class of bipartite graphs. In particular, we study Strong Bipartite Deletion, where given an undirected graph G and positive integers k and `, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G − S can be made bipartite by deleting at most ` edges. While fixed-parameter tractability when parameterizing by k or ` alone is unlikely, we show that this problem is fixed-parameter tractable (FPT) when parameterized by both k and `. Keywords and phrases fixed parameter tractable, bipartite-editing, recursive understanding 1 Introduction Graph-modification by either deleting vertices or deleting edges or adding edges such that the resulting graph satisfies certain properties or becomes a member of some well-understood graph class is one of the basic problems in graph theory and graph algorithms. However, most of these problems are NP–complete [19,27] and thus they are subjected to intensive study in algorithmic paradigms that are meant for coping with NP-completeness [9,10,22,23]. These paradigms among others include applying restrictions on inputs, approximation algorithms and parameterized complexity. The goal of this paper is to introduce a ‘stronger’ notion of graph deletion in the realm of parameterized complexity and illustrate the difficulties that arise when considering the family of bipartite graphs and provide an approach to obtain fixed-parameter tractability by overcoming these difficulties. A typical instance of parameterized graph deletion is of the following form. Let F be a family of graphs – such as edgeless graphs, forests, cluster graphs, chordal graphs, interval graphs, bipartite graphs, split graphs or planar graphs. The deletion problem corresponding to F is formally stated as follows. ∗ 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 © Ashutosh Rai, M. S. Ramanujan and Saket Saurabh; licensed under Creative Commons License CC-BY 33rd International Symposium on Theoretical Aspects of Computer Science (STACS 2016). Editors: Nicolas Ollinger and Heribert Vollmer; pp. 1–28 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 2 Strong Parameterized Deletion: Bipartite Graphs Figure 1 An example showing the strength of the new parameter for editing problems. F-Vertex (Edge) Deletion Parameters: k Input: An undirected graph G and a non-negative integer k. Question: Does there exist S ⊆ V (G) (or S ⊆ E(G)), such that |S| ≤ k and G − S is in F? In other words, given a graph G, can we delete at most k vertices or k edges such that the resulting graph belongs to F? An algorithm for F-Vertex (Edge) Deletion that runs in time f(k) · |V (G)|O(1) is called fixed-parameter tractable (FPT) algorithm and the problem itself is said to be FPT. The study of parameterized graph deletion problems together with their various restric- tions and generalizations has been an extremely active sub-area over the last few years. In fact, just over the course of the last couple of years there have been results on parameterized algorithms for Chordal Editing [5], Unit Vertex (Edge) Deletion [3], Interval Vertex (Edge) Deletion [4, 6], Planar F Deletion [9, 18], Planar Vertex Dele- tion [15], Block Graph Deletion [17] and Simultaneous Feedback Vertex Set [1]. Several known parameterized algorithms for F-Edge Deletion or the version where the objective is to delete k1 vertices and k2 edges utilize the fact that given a yes-instance (G, k) ∗ or (G, k1, k2) to the problem, there exists a vertex set S of V (G) of size k = k1 + k2 such that G − S∗ belongs to F. Clearly, this is true for any hereditary family of graphs; that is, if G ∈ F then all its induced subgraphs belong to F. Thus, if the corresponding F-Vertex Deletion is FPT then we can apply this algorithm and find a vertex subset S∗ of size at most k such that G − S∗ ∈ F. Having the set S∗ allows one to infer numerous structural properties of the input which can then be utilized in non-trivial ways to solve the original F-Vertex (Edge) Deletion problem. However, the existence of such a set is no longer guaranteed in the proposed stronger version of this problem. Let F be a polynomial-time recognizable family of graphs; that is, given a graph G, in polynomial time we can decide whether G belongs to F. For a fixed integer `, let F + `e denote the class of graphs that can be obtained by adding ` edges to a graph in F [2]. Furthermore, suppose that F-Edge Deletion is FPT with running time O(g(`) · nc). Here, n = |V (G)| and c is a fixed constant. That is, for any fixed integer `, F + `e can be recognized in time O(nc). The Strong F-Deletion problem is defined as follows. Strong F-Deletion Parameters: k1, k2 Input: An undirected graph G and non-negative integers k1 and k2. Question: Does there exist S1 ⊆ V (G) such that |S1| ≤ k1 and every connected component of G − S1 belongs to F + k2e? A. Rai, M. S. Ramanujan and S. Saurabh 3 A close inspection easily shows that Strong F-Deletion is much stronger than F- Deletion. For example, let F be the family of bipartite graphs and consider the graph n−1 G depicted in Figure 1. The graph G has 3 vertex disjoint triangles and the vertex v is adjacent to two vertices from each of the triangles. Clearly, after deleting v every connected component can be made bipartite by deleting exactly one edge. Thus, in the traditional n−1 sense this is a yes-instance of Bipartite Deletion with parameters (1, 3 ); however this is already a yes-instance of Strong Bipartite Deletion with parameters (1, 1). Thus, these problems seem much harder than traditional editing problems as the parameters are much smaller. An alternate viewpoint is that when G has a vertex set S of size at most k1 such that every connected component of G − S is in F + k2e then S can be considered a modulator into the graph class where every connected component of the graph belongs to F + k2e. While modulators to various polynomial-time recognizable graph classes have been studied in a very systematic way [8,9,14], the same is not true of modulators to NP-complete graph classes. Studying the Strong F-Deletion problem on the other hand allows us to do precisely this. In fact, it is not even necessary that the class F is a conventional graph class and instead can be the ‘composition’ of various graph classes. For instance F could be defined as the set of all graphs where each connected component is either chordal or a bipartite graph. The computation of such modulators is a problem with several algorithmic applications. For instance, in a recent work, Ganian et al. [11] used a similar notion in order to design algorithms for the classic Constraint Satisfaction Problem. In this article we study the Strong F-Deletion, when F is the family of bipartite graphs. Henceforth, F denotes the family of bipartite graphs. We call a graph G, t- pseudobipartite, if every connected component of G belongs to F + te. The problem we study is as follows. Strong Bipartite Deletion Parameters: k, ` Input: An undirected graph G and non-negative integers k and `. Question: Does there exist S ⊆ V (G) such that |S| ≤ k and every connected compon- ent of G − S belongs to F + `e? We refer to the set S as the solution for this instance. The primary reason behind the selection of the family of bipartite graphs for our study is that the problems where we are required to delete vertices and/or edges to obtain a bipartite graph are some of the most basic and well studied problems in parameterized complexity and studying these problems has led to the discovery of several new techniques and tools. These problems are called Odd Cycle Transversal and Edge Bipartization in literature and the algorithms with best dependence on the parameter have running time O(2.3146knO(1)) and O(1.977knO(1)), respectively [20, 25].
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