TY - JOUR

T1 - Simultaneous Feedback Edge Set

T2 - A Parameterized Perspective

AU - Panolan, Fahad

AU - Saurabh, Saket

AU - Zehavi, Meirav

AU - Agrawal, Akanksha

N1 - Funding Information:
A preliminary version of this paper appeared in the proceedings of the 27th International Symposium Algorithms and Computation (ISAAC 2016). The research leading to these results has received funding from the European Research Council (ERC) via grants Rigorous Theory of Preprocessing, reference 267959 and PARAPPROX, reference 306992.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/2/1

Y1 - 2021/2/1

N2 - 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 [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.

AB - 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 [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.

KW - Feedback edge set

KW - Parameterized complexity

KW - α-matroid parity

UR - http://www.scopus.com/inward/record.url?scp=85092375088&partnerID=8YFLogxK

U2 - 10.1007/s00453-020-00773-9

DO - 10.1007/s00453-020-00773-9

M3 - Article

AN - SCOPUS:85092375088

SN - 0178-4617

VL - 83

SP - 753

EP - 774

JO - Algorithmica

JF - Algorithmica

IS - 2

ER -