Improved FPT Algorithms for Deletion to Forest-Like Structures

The Feedback Vertex Set problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph G and a non-negative integer k, the objective is to test whether there exists a subset S ? V(G) of size at most k such that G-S is a forest. After a long line of improvement, recently, Li and Nederlof [SODA, 2020] designed a randomized algorithm for the problem running in time ?^?(2.7^k). In the Parameterized Complexity literature, several problems around Feedback Vertex Set have been studied. Some of these include Independent Feedback Vertex Set (where the set S should be an independent set in G), Almost Forest Deletion and Pseudoforest Deletion. In Pseudoforest Deletion, each connected component in G-S has at most one cycle in it. However, in Almost Forest Deletion, the input is a graph G and non-negative integers k,? ? ?, and the objective is to test whether there exists a vertex subset S of size at most k, such that G-S is ? edges away from a forest. In this paper, using the methodology of Li and Nederlof [SODA, 2020], we obtain the current fastest algorithms for all these problems. In particular we obtain following randomized algorithms. 1) Independent Feedback Vertex Set can be solved in time ?^?(2.7^k). 2) Pseudo Forest Deletion can be solved in time ?^?(2.85^k). 3) Almost Forest Deletion can be solved in ?^?(min{2.85^k ? 8.54^?, 2.7^k ? 36.61^?, 3^k ? 1.78^?})

Strong Parameterized Deletion: Bipartite Graphs

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 k and l and the objective is to check whether we can delete at most k vertices and l 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 k and l, 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 transformed into a graph in F by deleting at most l 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 l, 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 l edges. While fixed-parameter tractability when parameterizing by k or l alone is unlikely, we show that this problem is fixed-parameter tractable (FPT) when parameterized by both k and l

Parameterized Complexity of Deletion to Scattered Graph Classes

Graph-modification problems, where we add/delete a small number of vertices/edges to make the given graph to belong to a simpler graph class, is a well-studied optimization problem in all algorithmic paradigms including classical, approximation and parameterized complexity. Specifically, graph-deletion problems, where one needs to delete at most k vertices to place it in a given non-trivial hereditary (closed under induced subgraphs) graph class, captures several well-studied problems including Vertex Cover, Feedback Vertex Set, Odd Cycle Transveral, Cluster Vertex Deletion, and Perfect Deletion. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. While a precise characterization of the graph classes for which the problem is fixed-parameter tractable (FPT) is elusive, it has long been known that if the graph class is characterized by a finite set of forbidden graphs, then the problem is FPT. In this paper, we initiate a study of a natural variation of the problem of deletion to scattered graph classes where we need to delete at most k vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes. A simple hitting set based approach is no longer feasible even if each of the graph classes is characterized by finite forbidden sets. As our main result, we show that this problem (in the case where each graph class has a finite forbidden set) is fixed-parameter tractable by a O^*(2^(k^O(1))) algorithm, using a combination of the well-known techniques in parameterized complexity - iterative compression and important separators. Our approach follows closely that of a related problem in the context of satisfiability [Ganian, Ramanujan, Szeider, TAlg 2017], where one wants to find a small backdoor set so that the resulting CSP (constraint satisfaction problem) instance belongs to one of several easy instances of satisfiability. While we follow the main idea from this work, there are some challenges for our problem which we needed to overcome. When there are two graph classes with finite forbidden sets to get to, and if one of the forbidden sets has a path, then we show that the problem has a (better) singly exponential algorithm and a polynomial sized kernel. We also design an efficient FPT algorithm for a special case when one of the graph classes has an infinite forbidden set. Specifically, we give a O^*(4^k) algorithm to determine whether k vertices can be deleted from a given graph so that in the resulting graph, each connected component is a tree (the sparsest connected graph) or a clique (the densest connected graph)