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.