Simultaneous feedback vertex set: A parameterized perspective

For a family F of graphs, 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. For an integer α ≥ 1, an n-vertex (multi) graph G = (V, ∪_i=1 ^α Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colored (multi) graph. A natural extension of the F-Deletion problem to edge-colored graphs is the 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 G_i - S, where G_i = (V, E_i) and 1 ≤ i ≤ α, is in F. In this work, we study Simultaneous F-Deletion for F being the family of forests. In other words, we focus on the Simultaneous Feedback Vertex Set (SimFVS) problem. Algorithmically, we show that, like its classical counterpart, 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 2^O(αk)n^O(1) time and a kernel with O(αk^3(α+1)) vertices. The running time of our algorithm implies that SimFVS is FPT even when α ∈ o(log n). We complement this positive result by showing that if we allow α to be in O(log n), where n is the number of vertices in the input graph, then SimFVS becomes W[1]-hard. In particular, when α is roughly equal to c log n, for a non-zero positive constant c, the problem becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).