Simultaneous Feedback Edge Set: A Parameterized Perspective

In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G)→ 2[α], and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i ∈ [α], Gi − S is acyclic. Here, Gi = (V (G), {e ∈ E(G) | i ∈ col(e)}) and [α] = {1, . . . , α}. In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is same as the input of Sim-FVS and the objective is to check whether there is an edge subset S of cardinality at most k in G such that for all i ∈ [α], Gi − S is acyclic. Unlike the vertex variant of the problem, when α = 1, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for α = 3 Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic-graphs. The same reduction shows that the problem does not admit an algorithm of running time O(2o(k)nO(1)) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time O(2ωkα+α log knO(1)), where ω is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when α = 2. We also give a kernel for Sim-FES with (kα)O(α) vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph G, an integer q and, a coloring function col : E(G)→ 2[α]. The question is whether there is a edge subset F of cardinality at least q in G such that for all i ∈ [α], G[Fi] is acyclic. Here, Fi = {e ∈ F | i ∈ col(e)}. We give an FPT algorithm for Maximum Simultaneous Acyclic Subgraph running in time O(2ωqαnO(1)). All our algorithms are based on parameterized version of the Matroid Parity problem. 1998 ACM Subject Classification G.2.2 Graph Algorithms, I.1.2 Analysis of Algorithms