On the Parameterized Complexity of Simultaneous Deletion Problems

For a family of graphs F, an n-vertex graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A (multi) graph G = (V, cup_{i=1}^{alpha} E_{i}), where the edge set of G is partitioned into alpha color classes, is called an alpha-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_1, ldots, F_alpha)-Deletion problem. In the latter problem, we are given an alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i - S, where G_i = (V, E_i) and 1 leq i leq alpha, is in F_i. Recently, a subset of the authors considered the aforementioned problem with F_1 = ldots = F_alpha being the family of all forests. They showed that the problem is fixed-parameter tractable when parameterized by k and alpha, and can be solved in O(2^{O(alpha k)}n^{O(1)}) time. In this work, we initiate the investigation of the complexity of Simultaneous (F_1, ldots, F_alpha)-Deletion with different families of graphs. In the process, we obtain a complete characterization of the parameterized complexity of this problem when one or more of the F_i\u27s is the class of bipartite graphs and the rest (if any) are forests. We show that if F_1 is the family of all bipartite graphs and each of F_2 = F_3 = ldots = F_alpha is the family of all forests then the problem is fixed-parameter tractable parameterized by k and alpha. However, even when F_1 and F_2 are both the family of all bipartite graphs, then the Simultaneous (F_1, F_2)-Deletion} problem itself is already W[1]-hard

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^?})