Simultaneous Feedback Edge Set: A Parameterized Perspective

Abstract

Agrawal et al. (ACM Trans Comput Theory 10(4):18:1–18:25, 2018. https://doi.org/10.1145/3265027) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). Here, we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is an n-vertex graph G, a positive integer k, and a coloring function col:$E(G) \rightarrow 2^{[\alpha ]}$, and the objective is to check whether there is an edge subset S of cardinality k in G such that for each $i \in [\alpha ]$, $G_i - S$ is acyclic. Unlike the vertex variant of the problem, when $\alpha =1$, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for $\alpha =3$, Sim-FES is NP-hard, and does not admit an algorithm of running time $2^{o(k)}n^{{{\mathcal {O}}}(1)}$ unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time $2^{\omega k \alpha +\alpha \log k} n^{{{\mathcal {O}}}(1)}$ where $\omega$ is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when $\alpha =2$. We also give a kernel for Sim-FES with $(k\alpha )^{{\mathcal {O}}(\alpha )}$ vertices. Finally, we consider a “dual” version of the problem called Maximum Simultaneous Acyclic Subgraph and give an FPT algorithm with running time $2^{\omega q \alpha }n^{{\mathcal {O}}(1)}$, where q is the number of edges in the output subgraph.