Simultaneous Feedback Edge Set: A Parameterized Perspective

• Published in: Algorithmica Vol. 83; no. 2; pp. 753 - 774
• Main Authors: Agrawal, Akanksha, Panolan, Fahad, Saurabh, Saket, Zehavi, Meirav
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2021; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Feedback; Graph theory; Mathematics of Computing; Multiplication; Polynomials; Theory of Computation; Vertex sets; matroid parity; Parameterized complexity; Feedback edge set
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-020-00773-9

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 F eedback V ertex S et problem, called S imultaneous F eedback V ertex S et (S im -FVS). Here, we consider the edge variant of the problem, namely, S imultaneous F eedback E dge S et (S im -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 ] , 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 , S im -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 S im -FES running in time 2 ω k α + α 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 S im -FES with ( k α ) O ( α ) vertices. Finally, we consider a “dual” version of the problem called M aximum S imultaneous A cyclic S ubgraph and give an FPT algorithm with running time 2 ω q α n O ( 1 ) , where q is the number of edges in the output subgraph.