Simultaneous Feedback Edge Set: A Parameterized Perspective

Fahad Panolan, Saket Saurabh, Meirav Zehavi, Akanksha Agrawal

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Agrawal et al. (ACM Trans Comput Theory 10(4):18:1–18:25, 2018. 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 [α], and the objective is to check whether there is an edge subset S of cardinality k in G such that for each 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, and does not admit an algorithm of running time 2 o(k)nO(1) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time 2 ωkα+αlogknO(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 a “dual” version of the problem called Maximum Simultaneous Acyclic Subgraph and give an FPT algorithm with running time 2 ωqαnO(1), where q is the number of edges in the output subgraph.

Original languageEnglish
Pages (from-to)753-774
Number of pages22
Issue number2
StatePublished - 1 Feb 2021


  • Feedback edge set
  • Parameterized complexity
  • α-matroid parity

ASJC Scopus subject areas

  • Computer Science (all)
  • Computer Science Applications
  • Applied Mathematics


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