## Abstract

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], G_{i} - S is acyclic. Here, G_{i} = (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], G_{i} - 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(2^{o(k)}n^{O(1)}) unless ETH fails. This hardness result is complimented by an FPT algorithm for SIM-FES running in time O(2^{ωkα+α log log k}n^{O(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 ACYCLICSUBGRAPH. 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[F_{i}] is acyclic. Here, F_{i} = {e ∈ F | i ∈ col(e)}. We give an FPT algorithm for MAXIMUM SIMULTANEOUS ACYCLIC SUBGRAPH running in time O(2^{ωqα}n^{O(1)}). All our algorithms are based on parameterized version of the MATROID PARITY problem.

Original language | American English |
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Title of host publication | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |

Editors | Seok-Hee Hong |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 5.1-5.13 |

ISBN (Electronic) | 9783959770262 |

DOIs | |

State | Published - 1 Dec 2016 |

Externally published | Yes |

Event | 27th International Symposium on Algorithms and Computation, ISAAC 2016 - Sydney, Australia Duration: 12 Dec 2016 → 14 Dec 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 64 |

### Conference

Conference | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |
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Country/Territory | Australia |

City | Sydney |

Period | 12/12/16 → 14/12/16 |

## Keywords

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

## All Science Journal Classification (ASJC) codes

- Software