Simultaneous Feedback Edge Set: A Parameterized Perspective

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) -> 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 is an element of [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(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(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)(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(O(1)), where q is the number of edges in the output subgraph.