Simultaneous Feedback Vertex Set: A Parameterized Perspective

For a family of graphs F , a 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 graph G = (V,∪i=1Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the α-Simultaneous F-Deletion problem. In the latter problem, we are given an α-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph Gi \ S, where Gi = (V,Ei) and 1 ≤ i ≤ α, is in F . In this work, we study α-Simultaneous F-Deletion for F being the family of forests. In other words, we focus on the α-Simultaneous Feedback Vertex Set (α-SimFVS) problem. Algorithmically, we show that, like its classical counterpart, α-SimFVS parameterized by k is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant α. In particular, we give an algorithm running in 2O(αk)nO(1) time and a kernel with O(αk3(α+1)) vertices. The running time of our algorithm implies that α-SimFVS is FPT even when α ∈ o(logn). We complement this positive result by showing that for α ∈ O(logn), where n is the number of vertices in the input graph, α-SimFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014). 1998 ACM Subject Classification G.2.2 Graph Algorithms, I.1.2 Analysis of Algorithms