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,∪^α_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 G_i \ S, where G_i = (V, E_i) 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 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 α-SimFVS is FPT even when α ∈ o(log n). We complement this positive result by showing that for α ∈ O(log n), 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).